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2 


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MAR    26   1931 
APR   28   1931 

MAY  i  2  193* 
'  APR  29  19ST 


THE   PRESENT  TEACHING  OF 
MATHEMATICS  IN  GERMANY 


By 
DAVID  EUGENE  SMITH 


WITH  THE  CO-OPERATION  OF  VARIOUS 
GRADUATE  STUDENTS 


STATE  NOEMAL  SCHOOL 

LOS  ANGELES. 


PUBLISHED  BY 

Srarhrrs  GJuUrrjr,  (Enlttmbia  'iluturnntg 

NEW  YORK  CITY 

1912 


CONTENTS 

Chapter  Page 

*   I.  GERMAN  vs.  AMERICAN  CONDITIONS      .      David  Eugene  Smith      i 

^  II.  EVOLUTION  OF  THE  REFORM  IN  GERMANY      .      Isidore  Skolnick     12 

III.  THE  SECONDARY  SCHOOLS  OF  HESSE  AND  BADEN        .         .        .18 

Miriam  E.  West 

IV.  THE  SECONDARY  SCHOOLS  OF  THE  HANSEATIC  STATES        .         .25 

Katharine  S.  Arnold  and  Ruth  Fitch  Cole 

V.  THE  SECONDARY  SCHOOLS  OF  WURTEMBERG    .     Isidore  Skolnick     32 

VI.  THE  SECONDARY  SCHOOLS  OF  BAVARIA        .        M.  J.  Leventhal     39 

VII.  THE   HIGHER   SCHOOLS   FOR   BOYS   IN    PRUSSIA        .  .44 

Robert  King  Atwell 
VIII.  THE  SECONDARY  SCHOOLS  OF  ELSASS  AND  LOTHRINGEN      .        .     50 

Maurice  Levine 
'"  IX.  MATHEMATICS  IN  GERMAN  TECHNICAL  SCHOOLS        ...     63 

Donald  T.  Page 

X.  THE  GERMAN  MIDDLE  TECHNICAL  SCHOOLS    .    Miriam  E.  West     66 
'•XI.  MATHEMATICS  IN  THE  GERMAN  SCHOOLS  OF  NAVIGATION        .     75 

Donald   T.  Page 

XII.  COMMERCIAL  PROBLEMS  IN  THE  HIGHER  SCHOOLS  OF  GERMANY     78 

W.  F.  Enteman 

XIII.  MATHEMATICS  IN  THE  TEXT-BOOKS  ON  PHYSICS  .  A.  T.  French     90 

XIV.  GOVERNMENT  EXAMINATIONS  IN  PRUSSIA  AND  THE  NORTH  GER- 

MAN STATES  .  Cilda  Langfitt  Smith  and  Katherine  Simpson  97 

XV.  DESCRIPTIVE    GEOMETRY   IN    THE   REALSCHULEN        .        .         .  106 

Louise  Eugenie  Harvey  and  Jessie  Mae  Reynolds 

XVI.  CONCLUSION    .                                           .        Eleanora  T.  Miller  121 


<SA  \\ 


TEACHERS  COLLEGE  RECORD 

Vol.  XIII  MARCH,  1912  No.  2 


PREFACE 

In  one  of  the  graduate  classes  of  Teachers  College  it  is  the 
custom  to  devote  part  of  each  year  to  a  study  of  educational 
problems  in  the  field  of  mathematics  in  other  countries.  As  a 
rule  the  students  in  this  class  are  teachers  of  some  experience. 
All  of  them  are  college  graduates  and  all  are  desirous  of  know- 
ing the  best  that  the  world  is  doing  in  the  teaching  of  their 
chosen  subject.  In  this  desire  they  are  encouraged  in  every  prac- 
tical way,  it  being  the  position  of  the  department  that  such 
knowledge  is  one  of  the  two  foundation  stones  upon  which  they 
must  build,  the  other  being  a  sound  knowledge  of  the  subject 
matter.  With  these  two  must,  of  course,  go  much  else, — a 
knowledge  of  how  education  has  come  to  be  what  it  is,  an  out- 
look into  modern  tendencies,  a  knowledge  of  the  laws  of  mind, 
and  so  on ;  but  without  a  knowledge  of  mathematics  and  a  knowl- 
edge of  how  it  has  been  and  is  being  taught  at  its  best,  all  the 
rest  lacks  application  and  develops  the  mere  experimenter  (and 
too  often  an.  opinionated  one)  in  this  important  field. 

It  happens  that  during  the  current  year  the  German  branch 
of  the  International  Commission  on  the  Teaching  of  Mathe- 
matics has  issued  more  reports  than  that  of  any  other  country. 
All  of  the  most  important  nations,  including  our  own,  are  at 
present  issuing  these  reports,  but  Germany  has  thus  far  excelled 
all  others  both  in  the  range  of  the  investigation  and  in  the 
promptness  with  which  the  results  have  been  published.  On 

-67]  I 


2  Teachers  College  Record  [68 

this  account  the  present  class  has  given  more  attention  to  mathe- 
matics in  Germany  than  to  that  in  other  countries,  although 
covering  a  considerable  range  of  work  in  France,  Holland,  Eng- 
land, Sweden,  Russia,  Italy,  America,  and  Austria.  The  results 
of  some  of  this  investigation  have  been  summarized  by  the  vari- 
ous members  of  the  class  and  constitute  the  body  of  this  work. 

It  has  been  impossible  to  allow  enough  space  to  the  various 
reports  to  permit  of  many  details  as  to  the  schools,  the  courses, 
the  pupils,  the  teaching  staff,  and  the  methods  employed.  Such 
details  could  hardly  be  looked  for  in  a  condensed  statement  of 
this  kind.  All  that  has  been  attempted  in  these  reports  is  a  bird's- 
eye  view  of  the  field,  and  this  is  all  that  could  be  expected.  The 
details  have  been  presented  and  discussed  in  the  class,  but  the 
reader  who  wishes  for  more  specific  information  will  have  to 
consult  the  publications  in  their  original  form.  These  are  all 
available  at  a  reasonable  price,  and  may,  like  those  of  other 
countries,  be  secured  by  addressing  Messrs.  Georg  et  Cie,  Edi- 
teurs,  Geneva,  Switzerland.  If,  through  their  circulation  and 
through  such  abstracts  as  those  which  make  up  this  publication, 
we  can  awaken  to  the  large  questions  in  the  teaching  of  mathe- 
matics, the  work  of  the  International  Commission  will  prove  most 
salutary.  To  get  away  from  such  narrow  ideas  as  that  mathe- 
matical problems  must  all  be  physical  problems,  and  that  geometry 
fails  except  as  it  relates  to  the  workshop;  to  divorce  ourselves 
from  the  misconceptions  that  a  country  like  Germany  or  France 
no  longer  teaches  algebra  and  geometry  as  separate  and  distinct 
subjects;  to  come  to  the  state  of  forming  opinions  only  after 
finding  what  others  are  doing, — these  are  some  of  the  fortunate 
results  that  may  be  anticipated  from  such  study. 

With  respect  to  Germany,  it  should  first  be  understood  that 
the  schools  of  that  country  are  doing  certain  kinds  of  work  that 
we  are  not  doing,  some  that  we  cannot  do  under  present  condi- 
tions, and  some  that  we  might  not  care  to  do  if  we  could.  On 
the  other  hand  they  are  doing  certain  work  that  we  wish  we 
might  do,  and  that,  in  due  time,  we  shall  probably  come  to  do 
as  well  as  it  is  being  done  there.  Some  of  the  reasons  for  the 
difference  between  the  work  in  Germany  and  that  in  our  country 
are  set  forth  in  Chapter  I  and  several  desirable  lines  of  improve- 


69]       The  Present  Teaching  of  Mathematics  in  Germany        3 

ment  in  our  own  work  will  be  suggested  in  the  chapters  that 
follow. 

It  will  be  observed  that  these  reports  enter  but  little  into  the 
description  of  the  schools  themselves.  The  reader  who  wishes 
to  obtain  information  of  this  kind,  including  a  statement  of 
the  methods  of  conducting  the  classes  in  the  various  types  of 
schools,  may  consult  the  excellent  little  volume  prepared  a  few 
years  ago  by  Professor  J.  W.  A.  Young,  "  Mathematics  in  the 
vSchools  of  Prussia"  (New  York,  Longmans). 

For  the  benefit  of  the  reader  who  is  unfamiliar  with  the  Ger- 
man school  system,  a  brief  statement  of  the  character  and  scope 
of  the  work  of  the  secondary  schools  is  necessary  before  begin- 
ning the  various  chapters.  The  three  leading  types  of  secondary 
schools  are  as  follows :  the  Gymnasium,  with  both  Latin  and 
Greek ;  the  Realgymnasium,  with  Latin  but  no  Greek ;  and  the 
Oberrealschule,  with  neither  Latin  nor  Greek.  The  course  ex- 
tends over  a  period  of  nine  years  and  the  pupil  must,  in  general, 
have  attained  the  age  of  nine  years  before  he  can  enter.  The 
classes  are  named  as  follows,  beginning  with  the  lowest;  Sexta 
(VI),  Quinta  (V),  Quarta  (IV),  Untertertia  (UIII),  Obertertia 
(OIII),  Untersecunda  (UII),  Obersecunda  (Oil),  Unterprima 
(UI),  and  Oberprima  (OI). 

There  is  another  group  of  schools  offering  a  six-year  course, 
the  work  corresponding  exactly  to  the  first  six  years  of  the  nine- 
year  schools.  These  are  the  Progymnasium,  the  Realprogym- 
nasium,  and  the  Realschule. 

None  of  these  institutions  corresponds  exactly  to  any  Amer- 
ican school  and  each  is  consequently  referred  to  in  the  subse- 
quent chapters  by  the  German  name. 

The  labor  of  editing  the  several  monographs  that  make  up  this 
volume  has  fallen  chiefly  upon  Miss  Eleanora  T.  Miller,  who 
also  wrote  the  concluding  chapter,  and  to  her,  as  well  as  to  the 
various  contributors,  are  due  the  thanks  of  the  department. 

DAVID  EUGENE  SMITH. 


CHAPTER  I 

2  7  S /o 
GERMAN  VERSUS  AMERICAN  CONDITIONS 

David    Eugene    Smith 

Before  taking  up  the  special  reports  on  mathematics  in  the 
schools  of  Germany,  it  is  well  to  consider  briefly  one  question 
that  constantly  occurs  to  American  teachers  who  seriously  seek, 
after  studying  the  subject,  to  improve  the  mathematics  taught  in 
our  country.  The  question  is  a  very  natural  one,  viz. :  Why  can 
we  not,  in  the  same  number  of  years,  cover  as  wide  a  field  of 
mathematics  as  the  Germans?  Their  Gymnasium,  for  example, 
has  a  nine-year  course,  following  a  minimum  of  three  years  in 
the  elementary  school.  Here  are  twelve  years,  often  lengthened 
to  thirteen,  however,  to  be  placed  alongside  the  time  needed  to 
complete  the  work  through  our  high  school.  And  yet,  in  some 
parts  of  Europe,  and  in  particular  of  Germany,  students  in  such 
schools  study  not  merely  algebra  and  geometry  as  with  us,  but 
also  trigonometry,  descriptive  geometry,  geometric  drawing,  a 
little  modern  geometry  of  a  simple  kind,  some  analytics,  some 
calculus,  and  a  fair  amount  of  mathematical  mechanics.  What 
we  wish  to  know  is,  are  they  really  doing  this  ?  and,  if  so,  what 
is  their  curriculum?  how  is  the  work  arranged?  and  why  are  we 
not  doing  as  well  ? 

The  question  as  to  what  they  are  doing  in  Germany  is  briefly 
answered  in  the  subsequent  chapters.  The  types  of  schools,  the 
courses  of  study,  the  general  range  of  work,  the  preparation 
of  teachers,  something  of  the  methods  employed,  and  the  nature 
of  the  problems,  are  all  set  forth  with  sufficient  detail  to  give 
a  fair  reply  to  the  question.  But  it  was  not  the  problem  set 
for  the  writers  to  consider  the  reasons  for  the  difference  between 


6  Teachers  College  Record  [72 

Germany  and  America  in  this  respect,  and  hence  a  brief  dis- 
cussion of  this  question  is  in  order  at  this  time. 

In  the  first  place,  the  climate  of  Germany  is  thought  to  allow 
for  a  longer  school  year  than  with  us.  How  much  of  this  is 
really  the  case,  and  how  much  is  thought  to  be  the  case  because 
of  our  traditions,  it  is  impossible  at  present  to  say.  It  is  a 
fact,  however,  that  the  average  school  year  is  longer  in  Germany 
than  in  America,  and  that  our  excessively  hot  weather  in  June 
and  September,  not  to  speak  of  July  and  August,  is  practically 
unknown  there.  Neither  winter  nor  summer  gives  such  extremes 
of  temperature  as  are  found  with  us,  and  on  the  whole  the 
climate  is  better  adapted  to  a  long,  steady  pull.  Whether  we 
make  up  for  it  by  the  energy  and  the  rapid  work  of  our  stimu- 
lating winters  cannot  be  told.  At  any  rate,  as  to  the  school 
year,  they  have  the  advantage. 

The  same  is  true  as  to  the  school  day.  The  Germans  go  to 
school  earlier  in  the  morning  and  on  the  average  they  spend 
more  hours  there  than  we  do.  They  make  up  for  the  confinement 
by  long  walks  with  their  teachers  on  half  holidays,  and  by 
healthier  play  than  is  found  in  most  American  schools.  The  ele- 
ment of  time  in  contact  with  the  teacher  is  therefore  one  with 
which  we  must  reckon. 

Then,  too,  we  are  compelled  to  admit  that  their  teachers  are, 
on  the  whole,  better  prepared  than  ours.  They  know  their 
subject  better,  and  in  their  probation  year  they  have  been  under 
better  guidance  and  more  severe  drill  than  we  are  giving  in 
this  country.  It  is  not  to  our  discredit  that  this  is  so,  for  America 
has  done  well  when  we  consider  our  problem.  Ours  is  a  coun- 
try where  the  natural  resources  have  created  great  wealth  within 
a  brief  period.  Young  men  have  naturally  and  properly  been 
attracted  into  commercial  fields  rather  than  into  the  financially 
unremunerative  profession  of  teaching.  Even  if  our  nation  had 
grown  in  numbers  through  its  own  natural  increase  it  would 
have  been  difficult  to  furnish  men  for  our  high  schools ;  but  with 
our  influx  of  a  million  immigrants  a  year  we  have  been  nearly 
overwhelmed  with  the  task  of  educating  our  youth.  Had  not 
our  women's  colleges  developed  as  they  have  in  the  past  gen- 
eration, we  should  have  been  in  a  sad  state  in  our  effort  to  secure 


73]       The  Present  Teaching  of  Mathematics  in  Germany        7 

teachers.  When,  therefore,  we  hear  that  our  mathematics  does 
not  measure  up  to  the  standard  because  of  our  great  supply 
of  women  teachers,  we  should  remember  that  if  it  had  not  been 
for  the  entry  of  women  into  the  field  of  teaching  we  should 
have  had  to  face  the  dire  disaster  of  illiteracy. 

Then,  too,  it  must  be  said  that  the  leaders  in  educational  matters 
in  Germany  have  more  scholarship  than  has  generally  been  the 
case  in  America,  and  that  they  have  naturally  tended  to  appre- 
ciate scholarship  more  highly.  With  us  the  attack  on  a  subject 
like  the  classics  has  generally  been  made  by  people  who  knew 
little  or  nothing  of  Latin  and  Greek,  and  who  have  sought  to 
abolish  what  they  did  not  understand,  rather  than  to  improve  its 
presentation.  The  same  is  true  to  some  degree  with  mathematics, 
and  this  accounts  for  some  of  our  tendency  to  make  the  subject 
merely  one  for  the  workshop. 

Nevertheless  the  greatest  defect  in  our  system  to-day,  as 
compared  with  Germany,  is  to  be  found  in  the  lack  of  sound 
mathematical  training  on  the  part  of  the  teachers  themselves. 
Principles  of  education  are  valuable,  psychology  has  come  to 
be  a  necessary  part  of  a  professional  equipment,  the  history  of 
education  must  be  studied  to  lead  teachers  away  from  the  un- 
fortunate attempts  of  the  school  anarchist, — but  underneath  all 
this  must  lie  a  solid  foundation  of  mathematical  knowledge  if 
the  teacher  is  to  lead  students  to  know  and  to  love  the  subject 
and  to  know  its  bearings  upon  life.  There  is  where  Germany 
has  the  advantage,  and  to  the  lessening  of  this  advantage  those 
who  desire  the  best  for  the  American  schools  must  address  them- 
selves rather  than  to  an  aimless  iconoclasm. 

A  further  advantage  that  the  German  school  has  is  found  in 
its  general  plan.  Our  elementary  school  usually  covers  eight 
years,  the  teaching  being  done  by  women  who  have  rarely  had 
college  training,  and  who,  in  spite  of  all  their  devotion  to  the 
work,  cannot  have  a  broad  mathematical  outlook.  The  pupil 
then  enters  the  high  school,  and  usually  comes  under  the  instruc- 
tion of  a  teacher  (and  often  again  a  woman)  who  has,  at  the 
most,  studied  the  calculus.  The  mathematical  outlook  is  now 
better,  but  it  is  rarely  satisfactory.  But  there  is  a  decided  break 
in  the  pupil's  work  after  leaving  the  elementary  school.  In 


8  Teachers  College  Record  [74 

Germany  there  is  no  such  break,  for  the  Gymnasium  extends 
over  nine  years.  For  all  this  period  the  pupil  lives  in  the  same 
general  atmosphere,  with  the  same  corps  of  teachers.  This  is 
the  reason  why  Germany  can  successfully  begin  algebra  and 
geometry  earlier  than  we  can,  and  run  them  in  unbroken  sequence 
for  several  years.  If  we  begin  algebra  in  the  eighth  grade 
we  do  so  with  an  instructor  who  is  trained  solely  for  elementary 
work,  and  who  has  forgotten  most  of  the  algebra  she  once  knew ; 
but  in  Germany  they  can  begin  it  in  the  seventh  grade  with  the 
same  teacher  who  is  afterwards  to  present  the  trigonometry, — 
one  who  knows  mathematics  thoroughly.  Until  we  break  away 
from  our  idea  of  a  four-year  high  school  we  cannot  hope  for 
the  "  long  pull  and  strong  pull "  of  the  German  schools. 

The  reader  will  look  in  vain  in  the  German  reports  for  evi- 
dence of  the  hysterical  search  after  a  plan  for  studying  and 
benefiting  from  mathematics  without  any  work,  a  plan  that  seems 
to  be  in  evidence  in  some  parts  of  this  country.  Germany  recog- 
nizes fully  the  varied  capabilities  of  children,  and  of  course  the 
Gymnasium  is  only  one  type  (and  a  limited  one)  of  school. 
This  recognition  leads  to  a  greater  number  of  types  than  we  have 
developed,  although  we  seem  to  be  on  the  right  road  in  building 
up  our  various  forms  of  industrial  high  schools.  But  nowhere 
in  Germany  does  there  seem  to  have  developed  that  kind  of  mind 
that  seems  to  seek,  in  this  country,  to  abolish  algebra  and 
geometry  from  a  course  of  study  designed  to  give  an  all-round 
training.  Vocational  mathematics  there  is,  but  it  is  serious,  and 
in  general  it  recognizes  not  alone  the  immediately  practical  but, 
what  is  far  more  important,  the  potentially  practical  as  well. 
The  tendency  to  pay  the  most  attention  to  the  less  intellectual 
type  of  mind  is  rather  marked  in  America  to-day.  The  best 
thought  should  surely  go  as  much  to  the  encouragement  of  the 
boy  who  wishes  to  develop  intellectually  as  to  the  one  who  does 
not  wish  to  do  so.  In  this  country  to-day  one  has  to  search  not 
a  little  to  find  a  teachers'  association  that  is  discussing  the 
problem  of  teaching  more  mathematics  to  the  boy  who  wants 
to  learn,  but  he  has  no  trouble  in  hearing  all  sorts  of  plans 
for  teaching  no  mathematics  at  all  to  the  one  who  seems  ex- 


75]       The  Present  Teaching  of  Mathematics  in  Germany        9 

ceedingly  anxious  to  learn  nothing  that  makes  for  intellectual 
advance  and  for  higher  ideals. 

The  contrast  between  German  and  American  schools  is  not, 
however,  all  to  the  advantage  of  Germany.  It  is  by  no  means 
certain  that  a  smattering  of  analytics  and  the  calculus  would  be 
a  good  thing  for  our  pupils,  nor  that  it  is  a  good  thing  for 
theirs.  While  it  would  probably  be  a  desirable  plan  to  run  our 
algebra  and  geometry  over  a  longer  period  if  we  had  a  six- 
year  high  school  with  departmental  teaching,  and  possibly  side 
by  side  (although  there  are  other  factors  to  be  considered  which 
lack  of  space  prevents  mentioning  in  this  chapter),  still  there  is 
an  advantage  in  concentrating  the  attention  upon  a  subject  for 
a  briefer  time.  Our  course  in  algebra  is  in  many  respects  better 
than  the  German  course,  and  our  geometry  is  quite  as  thorough 
as  theirs  in  the  field  covered,  although  nobody  recognizes  better 
than  we  that  there  is  room  for  improvement.  With  the  present 
four-year  plan,  with  the  present  training  of  our  teachers,  and 
particularly  with  the  time  now  allotted  to  mathematics,  to  attempt 
to  cover  more  ground,  save  as  we  may  offer  electives  in  the  few 
schools  that  are  prepared  for  such  work,  would  be  a  decided 
step  backwards.  The  work  that  we  are  doing  is  by  no  means 
bad  work;  we  have  exceptionally  good  courses  in  algebra  and 
geometry,  and  these  we  are  constantly  seeking  to  improve  by 
making  them  appeal  more  to  the  interests  of  the  pupils,  and  by 
eliminating  material  that  we  cannot  justify.  We  usually  offer 
at  least  one  year  of  elective  work  for  pupils  who  are  mentally 
fitted  to  undertake  it,  and  shall  probably  offer  more  in  our  best 
schools  in  the  near  future.  Our  work  is,  therefore,  far  from 
being  all  bad,  nor  need  we  be  at  all  ashamed  of  what  our  better 
schools  are  doing,  or  of  what  our  system  as  a  whole  is  accom- 
plishing when  all  conditions  are  considered.  The  efforts  put 
forth  to  make  mathematics  interesting  and  vita!  are  apparently 
quite  as  good  in  America  as  in  Germany,  and  while  we  have  a 
great  deal  more  to  do  in  order  to  reach  an  ideal  presentation  of 
these  subjects,  the  spirit  shown  by  our  teachers  in  this  respect 
is  excellent.  Moreover  we  have  a  number  of  scientifically  con- 
structed text-books  in  arithmetic,  algebra,  and  geometry,  as  good 
indeed  as  can  be  found  anywhere,  considering  our  special  needs. 


io  Teachers  College  Record  [76 

These,  again,  must  continually  be  improved,  particularly  in  the 
direction  of  furnishing  a  motive  for  the  work  for  certain  types 
of  mind,  but  this  improvement  is  proceeding  in  a  creditable 
fashion.  We  have,  therefore,  not  a  little  that  stands  to  our 
credit  when  we  come  to  write  up  our  accounts.  So  true  is  this 
that  it  would  be  a  foolhardy  policy  to  attempt  any  wholesale 
destruction  of  our  present  courses  in  algebra  and  geometry,  or 
any  hasty  modification  of  our  curriculum. 

The  German  reports  should,  however,  convince  us  of  several 
things: 

1.  We  need  to  bend  every  effort  to  give  to  our  prospective 
teachers  a  better  knowledge  of  mathematics. 

2.  We  then  need  to  consider  carefully  the  possibility  of  extend- 
ing our  present  high-school  mathematics  over  a  longer  period 
with  the  same  corps  of  teachers;  in  other  words,  to  consider 
the  advantages  of  a  six-year  high  school. 

3.  We  also  need  to  get  ready  to  offer  higher  electives,  including 
trigonometry  and  its  applications,  geometric  drawing  (which  we 
are  entirely  neglecting),  a  form  of  practical  (and  potentially  prac- 
tical)   mathematics  that   is  not  merely  a  little   arithmetic,  and 
possibly  enough  of  the  calculus  for  simple  work  in  mechanics, 
and  enough  of  analytics   for  appreciating  this  amount   of  the 
calculus.     These  should  be  made  worth  the  while  for  the  stu- 
dent who  will  never  go  to  college,  and  they  can  surely  be  pre- 
sented in  such  way  as  not  to  harm  the  one  who  is  going  to 
pursue  his  studies  further. 

4.  We  also  need  to  set  our  faces  against  mathematical  work 
that  is  scrappy  and  without  scientific  content.    Work  of  this  kind 
is  not  found  in  such  countries  as  Germany, — countries  that  lead 
in  commerce,  in  agriculture,  and  in  industry,  as  well  in  educa- 
tional matters.     Germany  has  not  secured  for  mathematics  the 
status  that  it  has  in  all  her  schools  by  yielding  to  the  demands 
for  weak  algebra  and  weak  geometry,  or  for  none  at  all,  on  the 
part  of  men  who  know  nothing  of  the  subject  save  what  they 
got  from  a  poor  course  in  some  high  school.     Nor  shall  we 
make  a  worthy  place  for  the  subject  in  our  curriculum  by  con- 
sidering only  the  weakest  minds,  only  the  immediately  practical, 
and  only  that  which  any  untrained  teacher  can  present. 


77]       The  Present  Teaching  of  Mathematics  in  Germany      n 

5.  With  respect  to  vocational  training  we  must  recognize  that 
a  new  type  of  mind  has  appeared  in  our  high  schools,  demanding 
a  new  type  of  mathematics.    But  that  this  must  be  weak  mathe- 
matics, or  only  the  immediately  usable,  is  not  the  experience  of 
the  nation  that  has  made  the  greatest  strides  of  all  in  industrial 
work  in  the  past  generation.    We  may  well  take  counsel  of  that 
nation  in  training  our  future  artisans  in  the  power  of  thinking 
clearly  and  logically  along  mathematical  lines. 

6.  And  finally,  there  is  great  necessity  for  continued  advance 
in  the  training  of  leaders  among  the  teachers  of  our  country. 
The  average  college  cannot  do  this  work  under  present  condi- 
tions.   Indeed  the  feeling  is  still  prevalent  in  many  colleges  that 
professional  training  is  unnecessary.    One  of  the  most  important 
lessons  that  Germany  teaches  us  is  the  fallacy  of  this  position. 
The  establishing  of  strong  courses  for  teachers   in  the  senior 
year  of  our  colleges  will  do  much  to  improve  our  educational 
work,  particularly  in  our  secondary  schools. 


CHAPTER  II 
EVOLUTION  OF  THE  REFORM  IN  GERMANY1 

Isidore  Skolnick 

The  report  by  Dr.  Schimmack,  referred  to  in  the  footnote,  is 
divided  into  two  sections ;  the  first  dealing  with  the  reform  and 
its  progress  from  1840  to  1907,  and  the  second  dealing  with  the 
reform  and  its  progress  from  1907  to  the  present  day. 

Professor  Felix  Klein  in  the  introduction  to  this  report  refers 
to  a  statement  made  by  himself  at  Gottingen  in  March,  1911, 
that  for  some  time  he  had  wished  to  bring  before  the  public  the 
discussion  of  the  Commission,  and  to  make  known  his  own  per- 
sonal opinions  and  his  general  aim.  What  he  means  to  carry 
out  to-day  is  a  plan  which  he  has  heretofore  been  unable  to 
introduce.  The  idea  foremost  in  the  minds  of  many  teachers 
of  mathematics  in  Germany  at  present  is  that  the  concept  of 
"  function  "  as  defined  in  mathematical  language  should  be  the 
central  core  around  which  the  student's  knowledge  of  mathe- 
matics is  to  be  built.  The  purpose  in  introducing  the  idea  of 
function  is  not  to  teach  a  barren  analytic  geometry  but  to  make 
clear  the  simple  and  all-important  notion  that  upon  the  changing 
of  x  the  changing  of  y  depends.  It  is  expressly  stated  that  the 
function  idea  should  be  continually  borne  in  mind  in  the  teaching 
of  algebra  and  geometry.  It  is  urged  that  no  abstract  function 
idea  is  to  be  taught,  but,  on  the  contrary,  the  concrete  relations 
existing  between  x  and  y  as  functions  of  each  other  are  to  be 
expounded. 

It  may  be  of  interest  to  summarize  briefly  the  first  part  of 
the  report,  which  is  a  general  survey  of  the  teaching  of  mathe- 
matics from  the  year  1840  to  1860.  During  this  period  neither 

1  Die  Entwicklung  der  Mathematischen  Unterrichts  Reform  in  Deutsch- 
land.  von  Dr.  Rud.  Schimmack,  Oberlehrer  am  Gymnasium  zu  Gottingen, 
Leipzig  und  Berlin.  191 1. 

12  [78 


79]       The  Present  Teaching  of  Mathematics  in  Germany      13 

the  idea  of  function  nor  the  graph  was  presented  in  the  schools 
of  Germany.  Dr.  Schimmack  remarks  that  it  seems  strange 
that  anyone  could  understand  analytic  geometry,  maxima  and 
minima,  or  the  calculus,  without  being  familiar  with  this  func- 
tion idea. 

In  order  that  this  reform  movement  may  be  traced  more  defi- 
nitely and  closely,  it  will  be  well  to  follow  the  various  changes 
that  have  taken  place  from  time  to  time  in  Prussia.  As  early 
as  1812,  in  one  of  the  Prussian  Gymnasien,  the  following  courses 
in  mathematics  were  given:  a  very  few  simple  computations  in 
logarithms;  the  elementary  geometry  of  Euclid  from  book  one 
through  book  six,  together  with  books  eleven  and  twelve;  and 
a  little  plane  trigonometry.  Later  on,  about  1834,  progressions 
and  permutations  and  combinations  were  added.  In  1835  spher- 
ical trigonometry  and  a  little  work  in  conies  were  also  found  in 
the  course. 

The  Lehrplan  for  the  Prussian  Gymnasium  from  1837  to  1856 
shows  that  in  general  there  was  no  change  from  that  of  1835. 
Up  to  1867  no  analytic  geometry,  maxima  and  minima,  or  the 
calculus  were  to  be  found  in  the  Lehrplan.  The  only  additional 
subjects  given  for  the  Realschulen  of  the  first  order  (those  of 
1867  which  later  became  the  Realgymnasia)  were  the  following: 
elements  of  descriptive  geometry,  a  little  work  in  conies,  and 
also  some  statics  and  mechanics.  The  instructor,  if  he  were 
capable,  tried  to  supply  some  further  work  in  analytics,  some 
differential  and  integral  calculus,  and  a  little  astronomy.  All 
of  the  above  was  given  in  the  schools,  of  Prussia,  Schleswig- 
Holstein,  Hanover,  and  Hesse-Nassau.  Outside  of  Prussia  the 
Lehrplan  was  no  better  except  that  here  and  there  the  idea  of 
function  was  beginning  to  creep  in.  On  the  whole,  much  less 
mathematics  was  to  be  found  outside  of  Prussia  than  in  Prussia 
itself. 

About  the  year  1859,  K.  H.  Schnellbach,  a  member  and  organ- 
izer of  the  Examiners'  Commission,  founded  and  conducted  a 
mathematical  pedagogical  seminar.  About  the  same  time  a  book 
appeared  on  the  methods  of  mathematical  teaching,  by  Mekler 
(Das  Mekler  Lehrbuch,  1859).  This  book  contains  problems  in 
planimetry,  stereometry,  and  trigonometry,  with  some  work  in 


14  Teachers  College  Record  [So 

physics  involving  functions.  The  author  gives  a  brief  discussion 
of  simple  curves  related  to  rectangular  co-ordinates.  He  makes 
use  of  the  function  in  maxima  and  minima,  of  graphs,  and 
of  the  elements  of  the  calculus.  Another  noteworthy  book, 
Baltzer's  "  Elemente  der  Mathematik,"  followed,  and  was  in 
great  demand  in  the  teachers'  seminar.  This  book  dealt  with 
algebra,  the  new  geometry,  analytics,  and  the  calculus.  The 
period  from  the  year  1870  to  1890  is  characterized  by  a  number 
of  important  changes.  For  example,  a  Schulkonference  met  in 
1873  and  recommended  that  analytic  geometry  as  well  as  the  cal- 
culus be  considered  as  courses  in  the  schools. 

From  the  Lehrplan  of  the  Prussian  Gymnasium  of  1882  it 
would  appear  that  mathematical  teaching  was  neither  advancing 
nor  going  backward.  It  was  urged  by  those  who  framed  it  that  as 
much  mathematics  should  be  taken  up  as  would  render  one 
able  to  understand  the  mathematics  of  geography,  to  deal  with 
conies  intelligently,  and,  as  far  as  possible,  with  differential 
quotients.  In  this  Lehrplan  very  little  of  analytics  and  spherical 
trigonometry  was  included. 

In  1884,  Professor  M.  Simon,  Oberlehrer  of  the  Lyzeurh  Gym- 
nasium at  Strassburg,  advanced  the  proposition  that  the  ele- 
ments of  algebra  should  be  a  preparation  for  the  theory  of  func- 
tions. About  the  same  time  Dr.  A.  Hofler  of  Vienna  suggested 
that  it  would  be  well  if  Cartesian  co-ordinates  were  taught  in 
Class  VII  of  the  Gymnasium.  To  show  more  clearly  what 
courses  he  urged,  he  gave  certain  curves  that  might  be  plotted 
in  this  grade: 

y2  =  ex,   x2  ±  y2  =  c,  ax2  ±  by2  =  c, 
xy  =  a,  y™  =  ax',  x^"  =  a. 

For  transcendental  curves  he  suggested  y  =  log  x  and  y  =  sin  x. 
It  was  his  idea  that  the  plotting  of  these  curves  would  give  a 
more  concrete  meaning  to  the  function.  For  the  same  purpose 
he  urged  the  presentation  of  the  temperature  curve  and  its 
application  to  physics.  Such  curves  as  those  of  mortality  were 
also  suggested.  He  asserted  that  the  plotting  of  the  above  curves 
might  easily  be  introduced  in  the  first  three  classes  of  the  Gym- 
nasium. 
The  years  from  1890  to  1910  are  marked  by  even  more  radical 


81]       The  Present  Teaching  of  Mathematics  in  Germany      15 

changes.  About  the  year  1890  the  question  arose  as  to  whether 
or  not  conies  should  be  taught  in  the  Gymnasium,  and  it  was 
discussed  in  a  gathering  .of  teachers  of  mathematics  in  January 
of  that  year,  with  the  result  that  it  was  decided  to  encourage 
the  teaching  of  analytics  in  the  Gymnasium. 

In  the  Prussian  curriculum  of  1892  it  was  agreed  that  in  the 
last  classes  of  the  Gymnasium  there  should  be  a  more  detailed 
study  of  rectangular  co-ordinates,  with  special  reference  to  conies. 
It  was  also  arranged  that  analytic  geometry  in  general  should  be 
a  required  subject  in  the  course  of  study.  Outside  of  Prussia, 
the  students  of  the  Gymriasien  seem  to  have  had  no  knowledge 
of  co-ordinates  or  the  function,  nor  does  anything  seem  to  have 
been  done  with  analytic  or  advanced  synthetic  geometry. 

In  the  Lehrplan  of  1901,  the  function  concept  was  given  a 
prominent  place  and  the  connection  between  physics  and  mathe- 
matics was  emphasized.  Dr.  E.  Catling,  Oberlehrer  of  the  Gym- 
nasium at  Gottingen,  was  among  those  who  expressed  very  posi- 
tive opinions  that  the  variable  and  the  function  should  be  used 
and  taught  in  algebra,  geometry,  trigonometry,  and  analytics. 
'  Through  ignorance  of  the  function  idea,"  he  asserted,  "  one 
makes  mathematics  hard  for  himself." 

It  was  in  1904  that  Professor  Klein  made  this  important  state- 
ment :  "  The  geometric  function  idea  will  stand  as  the  important 
or  central  point  of  mathematical  teaching  and  will  be  carried 
as  far  as  possible,  and  by  this  means  the  elements  of  the  dif- 
ferential and  integral  calculus  will  be  introduced  into  the  high 
schools  of  Germany."  The  influence  of  this  remark  has  been 
very  far  reaching. 

The  reforms  since  1907  have  been  most  important  and  are 
discussed  in  detail  in  the  second  part  of  the  report.  At  the 
beginning  of  that  year  considerable  discussion  arose  as  to 
the  relation  between  mathematics  and  the  sciences,  and  conse- 
quently a  commission  was  appointed  for  the  purpose  of  suggest- 
ing various  courses  in  these  two  lines  of  work.  This  commission 
decided  upon  making  the  courses  in  mathematics  and  sciences 
distinct,  and  accordingly  a  separation  between  these  subjects  was 
agreed  upon  for  the  high  school.  One  group  was  called  the 
physics-mathematics  group  and  the  other  the  chemistry-biolog- 


16  Teachers  College  Record  [82 

ical  group.  The  idea  of  this  separation  was  to  give  more  de- 
tailed courses  in  physics,  chemistry,  biology,  and  mathematics. 
This  commission  was  called  the  Commission  of  German  Natural- 
ists and  Physicians,  and  it  met  at  Dresden  in  September,  1907. 
It  also  met  at  Leipzig  in  1908,  and  at  this  time  discussed  the 
subject  more  critically. 

Another  commission  met  in  1908  to  decide  what  should  be 
done  with  regard  to  the  sciences  and  mathematics  in  the  Girls' 
High  School.  Dr.  A.  Gutzmer  drew  up  a  report  upon  the 
details  of  its  discussions.  A  little  later  another  report  appeared 
dealing  with  the  question  of  the  place  of  mathematics  and  sciences 
in  the  new  girls'  high  schools  of  Prussia.  At  one  of  the  meet- 
ings of  the  commission.  Dr.  E.  Hoppe,  of  Hamburg,  told  the 
assembly  that  the  teaching  of  mathematics  and  sciences  according 
to  the  reform  plan  was  proving  to  be  a  great  success.  He  said 
that  the  idea  of  the  function  was  given  to  the  students  in  the 
Obersekunda,  and  that  they  found  it  interesting  and  very  easy 
to  understand.  He  stated  to  the  committee  that  this  led  him  to 
extend  the  function  idea  in  the  humanistic  Gymnasium.  Dr. 
H.  Schotten,  a  successful  teacher  of  the  reformed  mathematics 
in  this  high  school,  also  showed  how  the  work  could  be  presented. 

The  remainder  of  the  report  by  Dr.  Schimmack  is  devoted  to 
a  discussion  of  the  work  of  the  International  Commission  and 
the  reports  of  the  various  sub-committees,  all  of  which  are 
reviewed  in  detail  in  the  following  chapters.  It  will  perhaps  be 
helpful  to  mention  one  or  two  facts  which  are  not  brought  out 
in  the  separate  reports. 

In  tracing  the  growth  of  the  Gymnasium  in  Prussia  from  1901 
to  1909,  we  notice  a  gradual  increase  in  the  number  of  students 
from  87,478  to  102,297.  In  the  Prussian  Realgymnasium  there 
is  an  increase  from  21,078  to  41,202.  In  the  Oberrealschule  the 
increase  has  been  from  14,800  to  34,735  during  the  same  time. 
These  statistics  show  that  the  increase  has  been  greatest  in  the 
Oberrealschulen.  In  Bavaria  alone  we  find  nine  such  schools, 
namely :  Augsburg,  Passau,  Reglaburg,  Bayreuth,  Kaiserslantern, 
Ludwigshofen  (Rhein),  Munich.  Number^,  and  Wiirzburg. 
These  schools  give  courses  in  general  educational  subjects  similar 
to  those  given  in  the  Gymnasium  and  the  Realgymnasium  of 


83]       The  Present  Teaching  of  Mathematics  in  Germany      17 

Prussia.  The  following  table  gives  the  number  of  hours  devoted 
to  mathematics  in  the  Oberrealschulen  of  Prussia,  Bavaria,  Sax- 
ony, and  Baden: 

LOWER  UPPER  LOWER  UPPER  LOWER  UPPER 
Classes 6  543  3          2  2          1          1 

f  Arith.  5 

PRUSSIA  \  Math.                 -566  5          5  5          5          5 

I  Math,  draw'g (2)       (2)        (2)       (2)       (2) 

f  Arith.  4431  -  ___ 

BAVARIA  \  Math.  24  5          5  5          5          5 

[Math,  draw'g     --1        1          (2+)    (2+) 

f  Arith.  4442  2          1 

SAXONY   j  Math.                  --24  4          5  6          6          6 

[  Math,  draw'g -  1  222 

f  Arith.  5  _  _       _  __  __ 

t,Ar          \  Math.                 -555  5          5  5          5          5 

BADEN      I  Math,  draw'g -  2222 

This  shows  that  mathematical  drawing  is  compulsory  in  Sax- 
ony, Baden,  and  Bavaria,  but  is  not  compulsory  in  Prussia. 

In  1905  a  new  Lehrplan  was  drawn  up  to  meet  the  more  recent 
demands  for  the  girls'  high  school.  In  1908  a  new  school 
was  established  which  admitted  girls  at  the  age  of  six.  After 
completing  this  course  they  could  enter  the  Lyzeum  or  Frauen- 
schule,  the  latter  containing  also  a  teachers'  seminar  for  women. 
In  this  high  school,  mathematics  is  an  important  subject  and  is 
given  three  hours  a  week  throughout  the  course.  In  Prussia 
and  Saxony  the  course  followed  is  that  indicated  by  the  Prussian 
Lehrplan  of  1908.  This  is  a  ten-year  course  which  the  girls 
begin  at  the  age  of  six. 

Dr.  Schimmack's  report  also  gives  a  list  of  books  and  their 
titles  for  the  use  of  the  secondary  schools.  Among  the  books 
mentioned  are  Behrenden  and  Gotting's  "  Lehrbuch,"  P.  Crantz's 
"  Book  for  Girls'  High  Schools,"  and  H.  Dressler's  "  Study  of 
the  Function."  There  is  also  given  a  list  of  books  for  the  use 
of  the  instructor.  This  includes  such  books  as  E.  Borel's  "  Ele- 
ments of  Mathematics,"  J.  Druxes'  "  Book  on  Arithmetic  and 
Mathematics,"  F.  Klein's  "  Mathematical  Teaching  in  High 
Schools,"  and  "  Elementary  Mathematics  for  the  High  School," 
and  O.  Lesser's  "  Graphs  in  the  Mathematics  for  High  Schools." 


CHAPTER  III 
SECONDARY  SCHOOLS  OF  HESSE  AND  BADEN 

Miriam  E.  West 

The  two  reports  that  are  reviewed  in  this  chapter1  treat  of  the 
organization,  the  curriculum,  and  the  methods  of  mathematical 
instruction  in  the  higher  schools  of  Hesse  and  Baden.  The 
schools  of  Hesse  and  Baden  follow  in  general  the  plan  of  or- 
ganization of  the  higher  schools  throughout  Germany,  the  one 
exception  being  that  in  Baden  three  and  one-half  instead  of 
three  years  are  required  in  preparation  for  entrance. 

HESSE 

In  the  Grandduchy  of  Hesse,  a  country  with  a  population  of 
about  one  hundred  and  twenty-six  thousands,  there  are  the  fol- 
lowing schools:  ii  Gymnasien,  3  Realgymnasien,  7  Oberreal- 
schulen,  9  Realschulen,  5  higher  girls'  schools,  and  33  higher 
Burgerschulen.  These  last,  which  are  for  the  most  part  five-class 
schools  of  the  type  of  the  Realschule,  are  found  in  small  places. 
A  few  have  a  complete  seven-year  course.  There  are  several 
of  these  exclusively  for  girls.  The  higher  girls'  schools  are  ten- 
class  schools  admitting  girls  at  six  years  of  age.  In  connection 
Vvith  some  of  these  there  is  an  additional  course  of  three  or  four 
years  for  the  training  of  teachers. 

In  the  lowest  classes  of  the  girls'  schools  the  fundamental 
operations  are  taught,  and  are  followed  in  the  fifth  year  by 

1  Der  Mathematische  Unterricht  an  den  Hoheren  Schulen  nach  Organisa- 
tion, Lehrstoff  und  Lehrverfahren  und  die  Ausbildung  der  Lehramtskandi- 
daten  im  Grossherzogtum  Hessen,  von  Professor  Dr.  Heinrich  Schnell, 
Oberlehrer  am  Ludwig-Georgs-Gymnasium  in  Darmstadt.  April,  1910. 
_  Der  Mathematische  Unterricht  an  den  Hoheren  Schulen  nach  Organisa- 
tion, und  Lehrverfahren  und  die  Ausbildung  der  Lehramtskandidaten  im 
Grossherzogtum  Baden,  von  Hans  Cramer,  Professor  an  der  Goethe- 
schulc  in  Karlsruhe.  July,  1910. 

18  [84 


85]       The  Present  Teaching  of  Mathematics  in  Germany      19 

commercial  measures  and  the  subject  of  fractions,  which  is 
completed  in  the  sixth  year.  In  the  last  four  years  business  arith- 
metic is  studied,  and  in  the  ninth  and  tenth  years  some  geometry. 
In  the  ninth  year  the  fundamental  ideas  of  geometry  and  the 
computation  of  plane  figures  are  taken  up,  and  in  the  tenth  year 
the  computation  of  volumes  and  surfaces  of  solids.  In  the 
teachers'  training  course  this  work  is  reviewed  and  extended 
with  especial  emphasis  upon  rigorous  proof. 

Of  the  three  types  of  schools,  Gymnasium,  Realgymnasium, 
and  Oberrealschule,  the  Oberrealschule  devotes  the  greatest  num- 
ber of  hours  to  mathematics  and  the  Gymnasium  the  least.  The 
nine  classes  in  these  three  schools  are  numbered,  beginning  with 
the  lowest  class :  VI,  V,  IV,  Illb,  Ilia,  lib,  Ha,  Ib,  la.  One 
year  is  required  for  the  completion  of  each  class.  The  topics 
treated  in  the  various  classes  of  the  Realgymnasium  are  as 
follows : 


HRS. 
CLASS    PER 

WEEK 


SUBJECT 


VI  6  ARITHMETIC:  review  of  fundamental  operations  with  whole 
numbers,  abstract  and  denominate;  addition  and  subtrac- 
tion of  decimal  numbers;  factoring. 

V  4  ARITHMETIC:  common  fractions  with  concrete  illustrations; 
decimal  fractions. 

IV  5  ARITHMETIC:  common  and  decimal  fractions;  business  arith- 
metic; simple  rule  of  three. 

Plane   geometry   through    the    congruence    propositions   for 
triangles;  practice  in  use  of  ruler,  compasses,  squares,  etc. 

Ill  b       5         ARITHMETIC:  business  arithmetic  continued. 

Plane  geometry:  systematic  instruction  begun. 


Ilia 


II  a 


ALGEBRA:  through  powers  and  roots. 
Plane  geometry  continued. 


II  b        5         ALGEBRA:  logarithms;  equations  of  the  first  degree;  quad- 
ratic equations. 

Geometry:  computation  of  the  circle;  introduction  to  plane 
trigonometry. 


ALGEBRA:  quadratic  equations  with  more  than  one  unknown; 
interest  and  annuities. 


20 


Teachers  College  Record  [86 


HRS. 
CLASS    PER  SUBJECT 

WEEK 

I  b        5        TRIGONOMETRY  continued;  stereometry. 

ALGEBRA:  diophantine  equations;  combinations;  figurate 
numbers;  arithmetic  series  of  higher  order;  binomial 
theorem  for  positive  integral  exponents;  De  Moivre's 
theorem;  cubic  equations. 

STEREOMETRY  continued ;  elements  of  spherical  trigonometry. 

la         5        ALGEBRA:  determinants;  transcendental  series. 
ANALYTIC  GEOMETRY  of  straight  lines  and  conies. 

The  Oberrealschule  has  in  addition  to  these  a  course  in 
geometric-drawing  in  which  the  following  topics  are  considered : 

lib  and  Ha  CONSTRUCTION  of  plane  figures  and  oblique  geometrical  solids. 

Ib  DESCRIPTIVE  GEOMETRY:  review  and  extension  of  the  funda- 
mental propositions;  projection  of  solids;  plane  sections 
of  solids. 

la  REVIEW  :  solution  of  difficult  fundamental  problems  with 
representation  in  oblique  projection;  shadow  construction; 
elements  of  perspective;  solution  of  simple  practical 
problems;  right-angled  axonometry  and  oblique  parallel 
projection. 

The  arithmetic  as  given  offers  several  opportunities  for  an 
introduction  to  algebra.  In  the  review  of  the  fundamental  opera- 
tions in  the  first  year  the  pupil  is  made  familiar  with  parentheses. 
Fractions  are  considered  with  a  special  view  to  that  topic  in 
algebra.  The  rules  given  are  such  that  they  are  applicable  in 
algebra  as  well  as  in  arithmetic.  In  business  arithmetic,  by  the 
use  of  formulas  to  express  the  rules,  the  literal  symbols  of 
algebra  are  introduced. 

The  instruction  in  algebra  begins  with  the  concept  of  the  gen- 
eral number  for  which  the  way  has  been  prepared  in  arithmetic, 
if  the  mathematical  instruction  is  well  organized.  The  instruc- 
tion in  proportion  is  correlated  with  that  in  geometry.  There  is 
a  tendency  in  the  reform  movement  to  break  away  from  the 
old  Euclidean  type  of  proportion,  which  has  its  only  value  in 
geometry.  The  equating  of  the  products  of  the  terms  will 
answer  every  purpose  in  algebra.  Another  reform  has  been 
to  lessen  the  extent  to  which  formalism  rules  the  subject  matter 
in  algebra.  There  are  many  illustrations  of  this  formalism: 
for  instance,  the  division  of  long  polynomials,  a  process  which 


87]       The  Present  Teaching  of  Mathematics  in  Germany      21 

is  little  used  in  the  later  work;  in  the  subject  of  powers  and 
roots,  the  solution  of  problems  involving  high  degrees  and  com- 
plicated forms  which,  likewise,  are  of  little  value  in  later  work. 
The  problems  in  equations  as  well  as  many  of  those  in  interest 
and  annuities  are  not  practical.  In  many  schools  such  topics 
as  cube  root,  arithmetic  progressions  of  higher  order,  cubic  equa- 
tions, combinations,  and  diophantine  equations  are  omitted. 

At  present  little  use  is  made  of  the  graphic  representation  of 
functions.  In  the  schools  which  have  adopted  the  reform  plan 
the  graphic  representation  of  the  linear  function,  and  of  y  =  x2 
and  y  =  x3,  is  introduced.  Applications  are  found  in  physics  and 
in  the  solution  of  equations.  The  graphic  solution  of  the  quad- 
ratic of  one  unknown  quantity  is  not  practical.  By  the  use  of 
the  graphic  solution  of  the  cubic  equation  one  is  spared  the 
complicated  algebraic  method,  but  with  quadratic  equations  the 
graphic  method  is  not  easier  than  the  algebraic.  The  graphic 
treatment  is  of  value,  however,  in  quadratic  equations  of  two 
unknown  quantities  in  showing  the  existence  of  the  different 
roots.  Its  use  here  furnishes  an  excellent  introduction  to  analytic 
geometry. 

Plane  trigonometry  follows  the  course  in  geometry.  The  func- 
tions of  the  angles  and  computation  of  right  triangles  are  con- 
sidered. Logarithms  find  application  here,  but  many  teachers 
prefer  to  use  the  natural  value  of  the  function  first.  To  meet 
this  desire  the  following  plan  has  been  devised. 

lib    Trigonometry  limited  to  sine  and  cosine  proposition.     Stereometry 

limited  to  the  consideration  of  solids. 
Ha    General  stereometry,  goniometry,  and  logarithms. 

Where  this  plan  is  adopted  those  who  leave  school  at  the  close 
of  lib  will  have  no  unnecessary  logarithms  but  will  have  stereom- 
etry. Spherical  trigonometry,  which  is  taken  up  in  the  last  year, 
is  applied  to  mathematical  geography. 

Analytic  geometry  forms  the  conclusion  of  the  geometrical 
course.  In  it  the  study  of  both  branches  of  school  mathematics, 
algebra  and  geometry,  is  brought  to  a  close.  Some  teachers  use 
the  analytical  method  of  treatment  and  some  use  the  geometrical 
method,  while  many  combine  the  two  as  in  the  following  plan. 
The  sections  of  the  cone  and  the  focal  properties  of  the  curves 
are  considered  in  stereometry.  This  serves  to  introduce  the 


22  Teachers  College  Record  [88 

analytical  treatment,  in  which  problems  formerly  handled  graph- 
ically are  turned  around  by  asking  the  quertion,  whether  it  is 
possible  to  find  the  functions  of  the  different  curves,  having  given 
their  graphic  representation.  In  this  manner  the  different 
branches  of  mathematical  instruction  are  brought  together  anew. 
In  the  realistic  schools  the  pupils  are  able  at  the  end  of  the  course 
to  find  the  equations  for  simple  geometrical  loci. 

The  consideration  of  curves  of  all  kinds  leads  naturally  to 
the  idea  of  differential  quotients,  and  the  theory  of  maxima  and 
minima.  Thus  in  nearly  all  of  the  realistic  schools  some  work 
is  done  in  differential  calculus.  The  teachers  in  the  Gymnasien, 
however,  object  to  the  use  of  the  symbols,  although  they  do  not 
object  to  the  introduction  of  the  ideas  of  calculus.  With  the 
exception  of  the  study  of  the  history  of  mathematics  in  the  Ober- 
realschule  at  Heppenheim,  this  completes  the  instruction  in 
mathematics  given  in  the  higher  schools  of  Hesse. 

BADEN 

Besides  the  three  types  of  nine-class  schools  in  Baden  there 
are  a  few  six-  and  seven-class  schools.  The  general  manage- 
ment of  the  various  institutions  is  vested  in  the  state,  with  the 
exception  of  the  Realschulen  and  the  girls'  schools,  which  are 
partly  supported  by  the  community.  Certain  privileges  therefore 
concerning  selection  of  teachers  and  management  are  granted  to 
the  community.  For  entrance  to  any  of  these  schools  the  child 
must  pass  an  examination.  Girls  attend  the  higher  boys'  schools 
where  there  are  no  institutions  especially  for  them.  In  1909, 
8.2  per  cent  of  those  attending  these  schools  were  girls.  At  first 
only  the  Biirgerschulen  were  open  to  them,  but  since  1900  they 
have  been  admitted  to  the  other  schools.  The  greatest  number 
are  found  in  the  realistic  type  of  school. 

Some  of  the  distinguishing  features  of  this  report  are  as 
follows.  The  amount  of  mathematics  offered  in  the  schools 
of  Baden  does  not  differ  greatly  from  that  outlined  above  for 
those  in  Hesse.  In  the  girls'  schools  there  is  more  work  done 
in  mathematics  than  in  those  of  Hesse.  The  work  in  algebra 
includes  equations  of  the  first  degree  with  one  and  two  unknown 
quantities,  powers,  roots,  and  proportion. 


89]       The  Present  Teaching  of  Mathematics  in  Germany      23 

In  the  boys'  schools,  aside  from  slight  differences  in  the 
arrangement  of  topics,  there  are  three  ways  in  which  the  instruc- 
tion differs  from  that  in  Hesse:  (i)  The  intuitive  instruction 
given  in  geometry  in  the  Oberrealschulen  distinguishes  its  plan 
of  instruction  from  that  of  the  other  German  states.  (2)  Ac- 
cording to  the  report  the  use  of  models  is  much  more  ex- 
tensive. (3)  The  elements  of  differential  and  integral  calculus 
are  found  in  a  large  number  of  schools  of  all  three  types. 

The  intuitive  instruction  in  geometry  is  begun  in  class  V  and 
continued  through  three  years,  followed  in  class  Ilia  by  the 
formal  instruction.  In  the  first  year,  knowledge  is  gained  con- 
cerning the  various  plane  figures  and  their  properties  by  means 
of  looking  at  the  solids  and  plane  figures  as  well  as  by  drawing 
and  construction.  In  the  second  year  some  of  the  facts  con- 
cerning the  equality  of  plane  figures  are  observed  and  areas 
are  computed.  In  the  third  year  the  instruction  concerning 
solid  figures  proceeds  in  a  similar  manner.  If  this  instruction 
is  properly  carried  out  it  is  not  a  mere  diluted  form  of  the 
scientific  instruction,  nor  is  there  a  line  sharply  dividing  the 
two.  First  the  pupil  is  asked  to  state  the  results  of  his  observa- 
tion, and  after  several  observations  he  arrives  at  a  general  state- 
ment of  the  facts.  The  next  step  is  to  arrive  at  facts  that  are 
not  so  evident,  but  must  be  derived  by  the  aid  of  those  he 
already  has.  So  the  pupil  gradually  comes  to  feel  the  need  of  a 
proof.  It  is  a  method  which  makes  use  of  the  eye,  hand,  and 
mind  of  the  pupil. 

The  following  is  a  list  of  some  of  the  models  and  apparatus 
used :  for  elementary  arithmetic, — dry  measures  of  commercial 
forms,  cubic  decimeter  and  centimeter  of  wood  or  metal,  spheres 
and  circular  plates  in  whole  or  in  parts  for  fractions;  for  ele- 
mentary geometry, — wooden  models  of  all  geometric  forms,  a 
large  number  of  models  for  volumes  and  areas,  made  out  of 
wood  and  jointed ;  for  practical  trigonometry, — >all  the  various 
surveying  instruments.  The  recommendation  includes  models 
for  various  propositions  in  stereometry  and  spherical  trigonom- 
etry; sections  of  cones  and  cylinders,  the  cutting  plane  being  of 
different  material ;  spheres  with  zones  and  sectors  cut  out ;  models 
especially  designed  for  drawing;  and  the  slide  rule  for  the  use 
of  the  pupils. 


24  Teachers  College  Record  [90 

The  course  in  the  infinitesimal  calculus  includes  the  following : 
maxima  and  minima,  formation  of  infinite  reries,  determination 
of  lengths  of  arcs,  surface  area,  and  volume  of  simple  solids  of 
revolution.  Most  teachers  agree  that  the  concept  of  the  function 
and  the  graphic  representation  of  it  should  be  introduced  as 
early  as  class  III,  but  the  differential  quotients  and  integration 
should  be  left  for  the  last  two  years.  Problems  in  calculus  occur 
in  the  final  examinations  of  many  schools. 

The  final  examination  in  mathematics,  which  is  given  at  the 
completion  of  the  nine-years'  course,  is  both  oral  and  written. 
The  written  part  consists  of  four  problems,  two  from  algebra  and 
two  from  geometry.  These  are  selected  by  the  councilor  of  the 
state  board  from  problems  made  out  by  the  mathematics  teacher 
of  the  upper  class  of  the  institution  where  the  examination  is 
to  be  held.  The  problems  are  closely  related  to  the  mathematics 
of  the  last  school  year.  Besides  this  the  pupil  has  to  write  a 
theme,  the  subject  of  which  is  chosen  from  topics  considered 
in  the  upper  class. 

In  reading  these  two  reports  we  are  impressed  with  the  unity 
in  the  mathematics  curriculum,  a  fact  which  may  not  stand  out 
so  prominently  in  this  brief  review.  This  unity  is  accomplished, 
not  by  a  general  mixing  of  the  different  branches  of  mathematics, 
but  by  allowing  each  branch  to  serve  as  a  natural  introduction 
to  that  which  follows.  As  stated  before,  the  work  in  formulas 
in  arithmetic  prepares  the  way  for  the  literal  symbolism  of  alge- 
bra. The  graphic  work  in  algebra  and  the  study  of  conies  in 
stereometry  serve  as  an  introduction  to  analytic  geometry.  As 
soon  as  the  necessary  foundation  for  trigonometry  has  been  laid 
in  geometry,  that  subject  is  begun.  Logarithms  are  taken  up  in 
algebra  at  the  time  when  they  are  needed  for  trigonometry.  In 
Baden  the  preparation  for  the  study  of  calculus  is  begun  early 
in  the  course  by  the  introduction  of  the  concept  of  the  function 
and  its  graphical  representation.  The  fact  that  the  instruction 
in  mathematics  is  under  the  direction  of  special  mathematics 
teachers  for  at  least  six  years  makes  this  unity  to  a  large  degree 
possible.  We  may  well  afford  to  compare  this  with  the  isolated 
teaching  of  the  different  branches  of  mathematics  in  the  United 
States,  to  see  if  we  cannot  learn  a  lesson  from  Germany. 


CHAPTER  IV 

THE  SECONDARY  SCHOOLS  OF  THE  HANSEATIC 

STATES 

{Catherine  S.  Arnold  and  Ruth  Fitch  Cole 

The  report  on  mathematical  instruction  in  the  Gymnasien  and 
Realschulen  in  the  Hanseatic  states,  Mecklenburg  and  Olden- 
burg,1 was  prepared  from  the  results  of  investigations  carried  on 
by  sending  questionnaires  to  the  various  educational  centers.  The 
data  gathered  in  this  manner  gave  information  concerning  the 
curriculum,  the  methods  of  presentation  of  mathematical  material, 
and  the  topics  emphasized  in  the  teaching  of  the  various  subjects. 
Although  the  opinions  received  from  the  various  instructors  in 
mathematics  disagreed  in  a  few  details,  they  showed  certain 
general  tendencies,  which  are  embodied  in  this  report. 

The  course  of  study  in  a  Gymnasium,  a  Realgymnasium,  and 
an  Oberrealschule,  the  three  types  of  secondary  schools  in  Ger- 
many, is  given  in  order  to  show  what  topics  are  included  in  sec- 
ondary instruction  in  mathematics.  A  plan  from  a  reform  Gym- 
nasium is  also  added  for  the  purpose  of  indicating  the  influence 
of  the  new  movement,  the  so-called  reform  movement,  led  by 
Professor  F.  Klein. 

All  secondary  schools  have  a  nine  years'  course,  three  years 
being  required  for  entrance,  so  that  the  first  year  of  a  Gymnasium 
or  Real  institution  corresponds  to  the  fourth  grade  of  our  Amer- 
ican schools. 

The  following  table  gives  the  number  of  hours  per  week  de- 
voted to  mathematics  in  the  various  classes  of  the  three  types 
of  schools  in  Hamburg,  which  is  taken  as  a  representative  city. 

1  Der  Mathematische  Unterricht  an  den  Gymnasien  und  Realanstalten 
der  Hansestadte,  Mecklenburgs  und  Oldenburgs,  von  A.  Thaer  [Ham- 
burg], N.  Geuther  [Giistrow],  A.  Bottger  [Oldenburg],  Leipzig  and 
Berlin,  1911. 

91]  25 


26 


Teachers  College  Record 


[92 


Class1 

Gymnasium 

Realgymnasium 

Oberrealschule 


VI      V      IV  UIII  OIII  UII  OH     UI    01    Total 


4  33 

5  38 
5          46 


For  convenience  of  comparison,  the  original  Gymnasium  course 
and  the  suggested  reform  course  are  given  in  parallel  columns. 
This  work  begins  in  the  upper  third  class,  which  corresponds  to 
our  seventh  grade. 


GYMNASIUM 

— U 
Algebra,     through    linear    equations 

with  one  unknown. 
Geometry,  single      numerical      exer- 
cises with  plane  figures. 


Equations  with  more  than  one  un- 
known,— powers  and  roots. 


The  circle,  measurement  of  rectilinear 
figures,    similarity,    constructions. 


.REFORM  PLAN  OF  WILHELM 

GYMNASIUM 
Ill- 
Algebra,   linear    equations  and  fac- 
toring. 
Geometry,  the  circle. 

Ill— 

Equations  with  more  than  one  un- 
known,— geometric  construction  of 
irrational  quadratic  roots. 

Regular  polygons, — proportionality. 


— U  II— 


Quadratics,  irrational  quantities, — 
logarithms,  exponential  equations, 
— graphs  of  linear  and  quadratic 
functions. 

Measurement  of  polygons  and  circles. 


Equations  of  the  second  degree  with 
one  unknown,  powers  and  roots, 
logarithms. 


Proportionality  of 
monic  division, 
the  circle. 


the    circle, — bar- 
measurement    of 


— O  II— 


Powers  with  fractional  and  negative 
exponents,  progressions,  interest, 
and  income. 

Algebraic  geometry,  goniometry,  trig- 
onometry, beginnings  of  stereo- 
metry. 


Imaginaries, — exponential  equations, 
systematic  treatment  of  the  func- 
tion concept. 

Extension  of  geometric  representa- 
tion of  algebraic  expressions,  trig- 
onometry. 


— U  I— 


Plane  and  spherical  trigonometry,  go- 
niometry, stereometry,  analytic 
geometry  of  the  point,  line  and 
circle. 


Elements  of  spherical  trigonometry, 
arithmetical  and  geometrical  series 
of  the  first  order,  cubic  equations, 
stereometry. 


— O  I— 


Coordinates,  analytic  geometry  of  the 
conic  sections,  synthesis,  maxima 
and  minima,  map  projections,  re- 
view. 


Combinations,  probability,  binomial 
theorem,  analytic  geometry,  ele- 
ments of  differential  and  integral 
calculus. 


|  The  lowest  class  is  the  sixth,  then  the  fifth,  fourth,  lower  third,  upper 
third,  etcetera. 


93]       The  Present  Teaching  of  Mathematics  in  Germany      27 


Below  is  given  the  course  in  mathematics  for  two  types  of 
secondary  schools. 

REALGYMNASIUM  OBERREALSCHULE 

— U  Ill- 


Algebra  to  linear  equations  with  one 

unknown. 
The  circle,  constructions. 


Algebra  to  linear  equations  with  one 

unknown. 
Similarity  of  plane  figures. 


— O  Ill- 


Proportion,  powers  and  roots,  equa- 
tions of  first  degree  with  more  than 
one  unknown,  pure  quadratics. 

Similarity  of  plane  figures,  measure- 
ment of  rectilinear  figures,  construc- 
tions. 


Proportion,  powers  and  roots,  equa- 
tions of  first  degree  with  more  than 
one  unknown,  pure  quadratics. 

Proportionality  of  straight  lines  and 
circles,  polygons,  circumference  and 
area  of  the  circle. 


— U  II- 


Quadratic  equations,  fractional  and 
negative  exponents,  logarithms. 

Review  of  plane  geometry,  algebraic 
geometry,  projections,  elements  of 
trigonometry. 

— O 

Progressions,  exponential  equations, 
quadratic  equations. 

Harmonic  points,  rays,  poles, — simi- 
larity of  points  and  axes. 

Goniometry,  trigonometry  with  field- 
work,  stereometry  continued,  de- 
scriptive geometry,  spherical  trig- 
onometry and  applications. 


Quadratic  equations,  arithmetical  and 
geometrical  series  of  first  order, 
logarithms. 

Fundamentals  of  goniometry,  stere- 
ometry, elements  of  trigonometry. 

II— 

Trigonometry  and  stereometry  con- 
tinued, spherical  trigonometry  with 
applications,  quadratics. 


L 

Complex  numbers  and  their  geometri- 
cal representation,  cubic  equations, 
synthetic  treatment  of  conic  sec- 
tions, map  projections,  applications 
of  spherical  trigonometry  to  land, 
nautical,  and  astronomical  compu- 
tations. 


— U  I— 


Applications  of  algebra  to  geometry. 

Analytic  geometry  of  the  plane. 

Complex  numbers,  combinations,  bi- 
nomial, theorem,  cubic  equations, 
insurance. 


— O  I— 


Combinations,  probability,  binomial 
theorem,  differential  calculus,  max- 
ima and  minima,  indeterminate 
forms,  infinite  series,  analytic  geom- 
etry of  the  plane,  central  projec- 
tion. 


Differential  calculus,  maxima  and 
minima,  notion  of  continuity,  inde- 
terminate forms,  curves,  elements 
of  integral  calculus  with  applica- 
tions to  geometry  and  physics. 


Foremost  among  the  questions  which  have  occupied  the  atten- 
tion of  the  commission  is  that  of  the  best  place  in  the  course  of 
study  in  which  to  introduce  the  graphical  representation  of 
functions.  There  are  three  opinions  in  regard  to  the  matter: 


28  Teachers  College  Record  [94 

First,  that  the  fifth  year  is  not  too  early  for  the  elementary 
notions  of  functions;  second,  that  it  is  in  the  beginning  of 
trigonometry  that  the  subject  comes  most  naturally;  third,  that 
it  is  in  physics  that  the  most  advantageous  opportunity  for  its 
introduction  is  offered. 

Another  point  which  has  been  considered  is  the  subject  of 
drawing  in  connection  with  geometry.  Accurate  constructive 
drawing  is  considered  necessary  to  the  satisfactory  teaching  of 
mathematics.  In  some  schools,  neither  the  teachers  nor  the  pupils 
can  draw  properly.  The  difficulty  is  that  the  teachers  are  not 
trained  along  this  line,  and  it  is  evident  that  no  assistance  can 
be  given  by  the  teachers  of  drawing.  The  only  ones  who  can 
improve  the  situation  are  the  mathematics  teachers  themselves, 
who  should  be  skilled  in  drawing.  In  connection  with  courses 
in  stereometry,  in  several  schools  excellent  drawing  is  done,  and 
in  one  Gymnasium  working  drawings  are  constructed. 

The  instruction  in  geometry  has  also  been  found  unsatisfactory 
in  the  initial  presentation  of  the  subject  to  the  pupil.  Among 
the  suggestions  that  have  been  offered  are  the  following:  first, 
that  a  pre-geometric  course  be  introduced  leading  up  to  the  usual 
theorems  and  exercises;  second,  that  there  should  be  a  correla- 
tion of  like  propositions  in  plane  and  solid  geometry  (this  plan 
originated  in  Italy,  and  while  it  was  adopted  in  many  schools 
in  Germany,  it  is  no  longer  considered  with  favor)  ;  third,  that 
geometry  should  be  approached  from  the  standpoint  of  its 
practical  applications ;  and  fourth,  that  the  approach  on  the 
theoretical  side  is  preferable. 

As  a  further  means  of  vitalizing  mathematical  instruction,  a 
brief  historical  sketch  in  connection  with  the  several  topics  has 
been  suggested,  it  being  argued  that  methods  of  discovery  of 
the  fundamental  principles  of  mathematics  are  not  only  interest- 
ing but  necessary  for  the  student  to  know,  not  only  giving  him 
a  broader  outlook  but  also  suggesting  the  manner  of  approach 
for  future  original  research.  When  general  history  is  presented, 
why  not  associate  each  important  event  with  its  mathematical 
contemporary,  for  instance,  Hannibal  and  Archimedes,  the 
Reformation  and  the  cubic  equation,  the  Thirty  Years'  War  and 
analytic  geometry,  Napoleon  and  protective  geometry? 


95]       The  Present  Teaching  of  Mathematics  in  Germany      29 

Opinion  varies  as  to  the  manner  in  which  the  text-book  in 
mathematics  should  be  used.  While  some  maintain  that  the  book 
should  be  for  the  use  of  the  teacher  only,  and  thus  of  too  high 
a  standard  for  the  capabilities  of  the  pupil,  many  more  consider 
that  a  suitable  book  should  be  placed  in  the  hands  of  the  pupil. 
Some  express  themselves  as  desirous  of  a  manual  which  shall 
serve  as  a  mere  outline,  while  others  think  that  the  book  should 
contain  a  detailed  explanation  of  the  work.  In  general,  there 
seems  to  be  a  growing  opinion  that  the  pupil  should  have  access 
to  a  good  book  which  contains  the  demonstrations  and  which 
can  be  used  for  a  model.  Without  such  a  guide,  it  is  difficult 
for  him  to  construct  a  concise,  logical  proof  of  his  own. 

There  is  considerable  discussion  on  the  question  of  what 
apparatus  is  to  be  used  in  the  teaching  of  mathematics :  whether 
or  not  models  shall  be  in  the  hands  of  the  pupil,  and  if  so,  shall 
these  be  constructed  by  him.  There  are  times  when  a  model 
is  a  necessity,  for  example,  in  beginning  the  teaching  of  stereom- 
etry; and  the  blackboard  sphere  is  very  desirable  in  gaining 
the  concepts  of  spherical  trigonometry.  However,  the  excessive 
use  of  models  undoubtedly  dulls  the  imagination  of  the  pupil  and 
causes  him  to  lose  the  ability  to  visualize  the  propositions.  Yet, 
there  is  no  objection  to  placing  models  in  the  hands  of  the  pupil, 
provided  the  models  are  simple  ones,  constructed  of  such  ma- 
terials as  corks,  knitting  needles,  sticks,  or  clay.  If  the  pupil 
of  his  own  free  will  wishes  to  make  cardboard  models  at  home, 
he  should  be  encouraged ;  but  to  require  this  is  decidedly  un- 
profitable. Such  work  is  a  waste  of  time,  turns  mathematics 
into  handwork,  and  is  exceedingly  laborious  for  those  pupils 
who  are  not  skillful  with  their  hands.  Yet  there  are  some  very 
worthy  authorities  who  are  in  favor  of  having  the  pupil  perform 
this  construction  during  the  class  hour.  Undoubtedly  there  are 
instances  where  putting  together  a  model  brings  new  points 
to  the  notice  of  the  pupil.  On  the  other  hand,  apparatus  should 
not  stand  in  the  way  of  getting  a  clear  mathematical  insight  into 
the  problem ;  just  as  in  art  the  production  should  not  be  too 
realistic,  so  in  mathematics  there  must  be  some  work  left  for 
the  imagination.  At  the  end  of  this  discussion  an  extensive  list 
of  models  is  incorporated  in  the  report. 


30  Teachers  College  Record  [96 

There  is  a  general  agreement  upon  the  practical  applications 
in  mathematics.  Where  there  are  only  a  few  who  warn  us 
against  the  overuse  of  these  examples,  there  are  many  who  urge 
that  illustrations  be  taken  from  physics,  chemistry,  surveying, 
astronomy,  and  geography.  Some  claim  that  field  work  should 
occupy  two-thirds  of  the  time  allotted  to  mathematics.  Others 
are  of  the  opinion  that  there  is  too  large  a  number  of  students, 
too  great  a  scarcity  of  time  and  instruments,  and  too  few 
teachers  for  such  work ;  otherwise  it  would  be  most  profit- 
able to  have  much  practice  in  the  open  air.  Everywhere 
there  seems  to  be  a  demand  for  stronger  emphasis  on  the  appli- 
cations. The  teachers  assert  that  the  only  way  to  lead  to  the 
theoretical  is  by  means  of  the  practical,  while  the  patrons  are 
calling  for  work  of  practical  value.  This  method  of  teaching 
may  be  used  to  advantage  at  certain  stages  of  the  pupil's  progress, 
but  its  general  adoption  is  apt  to  frustrate  the  very  aims  and 
purposes  of  the  course  of  mathematics  in  the  secondary  schools. 
If  too  much  field  work  is  introduced  and  too  many  applications 
are  used,  there  is  danger  that  mathematical  instruction  may  pass 
entirely  into  the  domain  of  the  natural  sciences  and  that  the 
vigor  of  sound  mathematical  reasoning  be  lost. 

Agitation  for  the  separation  of  the  sciences  into  two  main 
groups,  namely,  a  mathematical-physical  group  and  a  chemical- 
biological  group,  is  current.  Many  teachers  of  mathematics 
are  in  favor  of  such  a  division.  As  our  courses  are  arranged 
to-day,  such  a  classification  is  not  natural  and  would  not  furnish 
adequate  preparation  for  the  courses  in  physics  and  chemistry  at 
the  universities.  A  valid  claim  might  also  be  made  that  mathe- 
matics is  related  to  geography,  mineralogy,  or  biology.  Applied 
mathematics  is  the  natural  companion  of  pure  mathematics  and 
should  be  kept  in  that  department. 

The  recommendations  of  the  commission  set  forth  in  this 
report  are  conservative,  yet  they  show  that  the  suggestions  of 
the  reformers  have  had  their  influence  and  have  served  to  enrich 
and  improve  mathematical  instruction.  Subjects  worthy  of  in- 
terest are  the  early  introduction  of  the  function  idea,  the  extent 
to  which  the  use  of  models  shall  be  carried,  and  the  use  of  the 
text-book.  We  find  that  in  Germany,  as  in  America,  a  general 


97]       The  Present  Teaching  of  Mathematics  in  Germany      31 

demand  for  practical  applications  is  voiced  by  all  interested, 
and  that  teachers  of  pure  mathematics  are  anxious  that  this 
demand  shall  not  spoil  the  presentation  of  the  subject.  In  gen- 
eral, the  report  shows  that  mathematical  instruction  in  Germany 
is  strong  in  every  particular;  that  the  demand  for  improvement 
is  carefully  discussed  by  educational  authorities;  and  that  the 
needs  are  very  similar  to  the  needs  of  mathematical  instruction 
in  America. 


CHAPTER  V 
THE  SECONDARY  SCHOOLS  OF  WURTTEMBERG1 

Isidore   Skolnick 

This  report  is  divided  into  seven  chapters.  The  first  chapter 
treats  of  the  various  schools  in  Wiirttemberg.  The  second  and 
third  chapters  give  an  account  of  the  mathematics  course  offered 
and  of  the  various  reforms  which  have  been  proposed  in  the 
Gymnasium  and  Realgymnasium  respectively.  The  fourth  re- 
views the  mathematics  of  the  Madchenschulen  and  hohere  Lehr- 
erinnenseminar  of  Stuttgart.  In  chapter  five  the  text-books  on 
arithmetic,  algebra,  geometry,  stereometry,  and  other  subjects 
are  reviewed.  The  sixth  chapter  gives  an  account  of  the  exam- 
inations, while  the  seventh  and  last  chapter  contains  facts  con- 
cerning the  preparation  of  teachers  for  the  high  schools. 

TYPES  OF  SCHOOLS 

The  three  types  of  schools,  the  Gymnasium,  the  Realgym- 
nasium, and  the  Oberrealschule,  are  found  in  the  kingdom  of 
Wiirttemberg  and  vicinity.  The  arrangement  of  classes  in  these 
schools  differs  slightly  from  that  of  the  North  German  schools 
and  the  comparison  is  given  in  the  table  which  follows. 


WURTTEMBERG 

NORTH  GEF 

fCb 

188           11 

Lower  division        •{ 

11 

Unterstufe 

1 

mj 

IV] 

Middle  division      ] 

V  > 

Mittelstufe 

[ 

VI  J 

c 

VII] 

Higher  division      •{ 

VIII  \ 

Oberstufe 

I 

IX  J 

1  Der  Mathematische  Unterricht  an  den  Hoheren  Schulen  nach  Organi- 
sation, Lehrstoff  und  Lehrverfahren  und  die  Ausbildung  der  Lehramts- 
kandidaten  im  Konigsreich  Wurttemberg,  von  Dr.  Erwin  Geek,  Leipzig 
und  Berlin,  1910. 

32  [98 


99 j       The  Present  Teaching  of  Mathematics  in  Germany      33 

Shortly  before  1903,  Class  I  was  divided  into  two  parts,  thus 
making  ten  classes  instead  of  nine.  The  average  age  of  the 
students  on  entering  the  first  class  is  eight  years,  which  means 
that  they  graduate  at  the  age  of  eighteen.  In  certain  sections  of 
Wurttemberg  there  are  elementary  schools  having  a  two-  and 
three-year  course. 

In  1787  the  first  Realschule  was  founded  in  Niirtingen,  and 
in  1793  additional  schools  of  this  type  were  recognized  by  the 
government.  The  Realschulen  are  attended  by  the  older  students, 
who  take  such  courses  as  will  prepare  them  to  become  practical 
business  men.  There  are  also  a  number  of  schools  that  give 
theological  courses. 

The  Biirgerschulen  of  Stuttgart  are  midway  between  the  Real- 
schulen and  the  Volkschulen  (common  schools).  They  differ 
from  the  Realschulen  in  that  English  is  elective,  that  there  are 
eight  classes  instead  of  ten,  and  that  the  lowest  class  may  be  en- 
tered by  pupils  who  have  attained  the  age  of  six  years.  They 
are  primarily  for  those  who  are  to  enter  the  commercial  field. 

The  high  schools  for  girls  have  ten  classes,  the  girls  entering 
at  the  age  of  six.  At  Stuttgart  there  is  a  higher  seminar  for 
girls  connected  with  their  high  school,  and,  in  addition  to  this, 
an  excellent  school  called  the  Madchengymnasium.  In  other 
localities,  where  there  are  no  high  schools  provided  for  the  pur- 
pose, girls  are  permitted  to  attend  the  higher  boys'  schools,  the 
Knabenschulen. 

A  statement  of  the  schools  of  Wurttemberg  tabulated  as  to 
types  and  number  is  here  given : 

A.  Humanistic  schools. 

1.  Gymnasien  (14)  and  Theological  Seminaries  (4). 

2.  Progymnasien  with  upper  classes  (5). 

3.  Landschulen  (48). 

B.  Realgymnasien. 

1.  Realgymnasien,  all  classes  (5). 

2.  Real  progymnasien,  with  two  upper  classes  (8). 

C.  Realschulen. 

1.  Oberrealschulen,  all  classes  (12). 

2.  Realschulen,  with  one  upper  class  (17);  with  two  upper  classes  (4). 

3.  Landschulen  (68). 

4.  Burgerschulen,  in  Stuttgart,  (2). 

D.  Madschenschulen,  public  (17);  private  (6). 

Teachers'  Seminar  for  Women  in  Stuttgart. 


34  Teachers  College  Record  [100 

THE  COURSE  IN  MATHEMATICS  IN  A  GYMNASIUM  PRIOR 
TO  THE  REFORMS  OF  1891 

Class  I  Addition,  subtraction,  multiplication,  and  division  by  a  two 
figure  divisor;  tables  of  measures,  weights,  and  time  with 
problems  involving  the  same. 

Class  II  Easy  decimals  and  a  little  percentage  with  occasionally  some 
work  in  proportion. 

Class      III    Factoring,  prime  factors  and  fractions. 

Class  IV-V     Fractions  continued,  including  complex  fractions,  and  discount. 

Class  VI  Algebra:  fundamental  operations,  simple  equations  of  the  first 
degree,  theory  of  indices  and,  proportion.  Geometry:  dis- 
cussion of  lines,  angles  and  triangles,  and  propositions  on  the 
parallelogram  and  circle,  the  teacher  dictating  the  proofs. 

Class  VII  Algebra:  solution  of  simple  equations  and  equations  of  the 
first  degree  in  more  than  one  unknown.  Geometry:  plane 
geometry  continued,  with  algebraic  interpretation.  Trig- 
onometry: fundamental  notions  of  trigonometric  functions 
and  some  formulas. 

Class  VIII  Algebra:  study  and  use  of  logarithms,  affected  quadratic  equa- 
tions, simultaneous  quadratic  equations,  and  theory  of 
experiments. 

Class      IX     Algebra:  binomial  theorem. 

MATHEMATICS  IN  THE  REALGYMNASIUM  AND  REALSCHULE 

About  the  year  1810  the  students  of  the  Stuttgart  Gymnasium 
expressed  dissatisfaction  with  the  existing  course  of  study,  claim- 
ing that  the  Latin  and  Greek  to  which  they  were  obliged  to 
devote  a  considerable  portion  of  their  time,  did  not  fit  them  for 
their  subsequent  positions  in  life.  They  were  to  become  officers 
and  "  Hofleute "  and  therefore  they  needed  a  more  extensive 
knowledge  of  mathematics  than  of  Greek.  In  response  to  this 
demand  the  curriculum  was  divided  into  groups  A  and  B,  group 
A  retaining  the  Latin  and  Greek  and  group  B  having  an  increased 
amount  of  mathematics  and  modern  language.  These  changes 
were  made  effective  in  classes  VI  and  VII  of  the  Obergymnasium. 
The  students  displayed  great  ability  and  enthusiasm  for  the  work 
in  modern  languages  and  mathematics,  and  in  1859  there  was  a 
special  teacher  giving  instruction  in  algebra,  geometry  and 
geometric  drawing. 

As  early  as  1865  the  need  for  men  trained  in  the  various 
lines  of  scientific  work  began  to  be  felt  very  keenly,  and  the 
outcome  of  it  was  a  clear  division  of  the  schools,  the  Realgym- 


loi]     The  Present  Teaching  of  Mathematics  in  Germany      35 

nasium  becoming  separate  and  distinct  from  the  old  Gymnasium, 
and  developing  into  a  scientific  school.  In  1871  the  first  teachers' 
examinations  were  held  in  that  school,  and  in  the  same  year  a 
new  program  was  formulated  which  offered  to  the  student  the 
desired  and  necessary  subjects  in  mathematics.  Mathematics 
was  entirely  eliminated  from  classes  VI  and  VII  and  in  classes 
VIII  and  IX  analytic  geometry,  spherical  trigonometry,  and 
mathematical  geography  were  given.  In  the  Stuttgart  Realgym- 
nasium,  the  proportion  of  the  students'  time  (expressed  in  the 
number  of  hours  out  of  the  total  number  spent  on  all  subjects) 
which  was  devoted  to  mathematics  in  the  various  classes  was  as 
follows:  I,  4  out  of  25;  II,  4  out  of  25;  III,  4  out  of  31; 
IV,  4  out  of  31 ;  V,  5  out  of  32;  VI,  5  out  of  33;  VII,  7  out 
of  33;  VIII,  17  out  of  34;  IX,  8  out  of  34. 

In  the  Stuttgart  Realschule  the  course  extended  over  four 
years.  The  purpose  of  the  school  was  to  prepare  men  for 
scientific  and  technical  work,  and  almost  one-half  of  the  students' 
time  was  devoted  to  work  in  mathematics.  This  work  included 
all  of  the  subjects  from  arithmetic  through  spherical  trigonometry. 

The  reforms  of  1904  and  1906  made  the  Oberrealschulen  of 
Wiirttemberg  true  technical  schools.  In  the  upper  classes  the 
number  of  hours  devoted  to  mathematics  was  decreased  and  the 
number  given  to  the  other  sciences  was  increased. 

THE  HIGHER  GIRLS'  SEMINAR 

The  purpose  of  the  Girls'  Seminar  in  Stuttgart  is  to  offer 
courses  to  girls  who  are  preparing  to  teach  in  the  lower  and 
middle  divisions  of  the  girls'  high  schools.  The  course  extends 
over  a  period  of  three  years  and  is  encyclopedic  in  nature,  with 
the  emphasis  put  upon  pedagogy.  The  work  of  the  first  two 
years  is  given  in  detail  later.  The  third  year's  work  is  mainly 
a  specific  preparation  for  what  is  called  the  "  leaving  examina- 
tion," the  successful  passing  of  which  entitles  the  candidate  to 
teach.  The  subjects  in  which  they  are  examined  are  religion, 
German,  French,  English,  history,  geography,  natural  history, 
mathematics,  hygiene  and  pedagogy. 

We  find  in  the  three  years'  course  in  mathematics  of  this  school 
an  interesting  attempt  to  give  a  general  survey  of  the  field  of  the 


36  Teachers  College  Record  [102 

less  advanced  secondary  mathematics.  It  presupposes  the  aver- 
age amount  of  preparation  in  elementary  mathematics  and  in- 
cludes work  in  algebra,  geometry,  trigonometry,  and  stereometry 
with  which  is  combined  geometric  drawing.  Although  the  Eucli- 
dean geometry  is  strictly  followed,  they  enter  into  discussions 
concerning  the  reorganization  of  the  subject  matter  as  to  the 
number  and  sequence  of  theorems.  Some  topics  as,  for  instance, 
inscribed  and  circumscribed  polygons  and  the  theory  of  limits, 
which  are  omitted  from  the  ordinary  girls'  high  school  course, 
are  given  here.  The  concept  of  function  and  the  graphic  repre- 
sentation of  algebraic  functions  claim  their  share  of  the  students' 
attention.  The  work  in  trigonometry  includes  the  solution  of  the 
right  triangle,  and  practical  problems  are  given  which  afford 
practice  in  the  use  of  four-place  logarithmic  tables.  Spherical 
trigonometry  is  not  emphasized.  Stereometry  is  touched  upon 
briefly,  the  work  being  confined  to  a  study  of  the  prism,  pyramid, 
cylinder  and  sphere.  The  texts  used  are  Speeker's  "  Geometry  '*' 
and  Bardy-Hartenstein's  "Algebra."  For  the  classes  in  trigo- 
nometry and  stereometry  there  are  no  text-books,  manuscripts 
being  used  instead.  A  very  little  work  in  physics,  supplemented 
by  just  enough  chemistry  to  make  the  physics  clear,  constitutes 
the  course  in  these  branches  of  science. 

At  the  termination  of  the  course  the  students  are  given  an 
examination  which  is  divided  into  two  parts.  The  first  part, 
given  at  the  end  of  the  second  year,  is  a  test  in  rapidity  as  well 
as  in  content.  The  second  part  determines  the  qualification  of 
the  candidate  for  high  school  positions. 

THE   ToCHTERSCHULEN 

The  Tochterschulen  are  not  strictly  secondary  schools  but 
there  is  a  seminar  in  connection  with  the  higher  classes  in  which 
secondary  subjects  are  taught.  These  schools  are  either  public 
or  private  and  contain  ten  classes.  In  the  lower  classes  instruc- 
tion is  offered  in  what  is  termed  biirgerlicher  Rechnen,  but  in 
the  upper  classes  or  the  seminar,  algebra  and  geometry  are  taught 
by  a  high-school  teacher. 

In  1903  the  public  school  board  decided  that  the  standard  of 
these  girls'  schools  should  be  raised,  and  accordingly  a  regular 


103]     The  Present  Teaching  of  Mathematics  in  Germany      37 

and  fairly  extensive  mathematics  course  was  incorporated  into 
the  school  curriculum.  The  complaint  which  one  hears  to-day  in 
connection  with  these  schools  is  that  the  teachers  are  of  the  old, 
non-progressive  type  and  that  they  are  slow  to  adopt  any  of  the 
reforms  in  mathematics  teaching.  Considerable  attention  is  paid 
to  the  work  in  arithmetic  which  includes  a  large  amount  of  busi- 
ness calculation,  four  hours  per  week  being  devoted  to  the  work 
in  the  first  five  classes,  three  hours  in  the  sixth  and  seventh, 
and  two  in  the  last  three  classes.  Special  emphasis  is  put 
upon  mental  work.  In  classes  IX  and  X,  three  hours  are  given 
to  geometry.  The  more  important  propositions  of  plane  geometry 
are  studied  and  work  in  geometric  drawings,  exercises  and  con- 
structions is  required.  The  course  in  algebra  is  not  extensive. 

MATHEMATICS  TEXT-BOOKS  IN  GERMANY 

"  Die  Methodischen  Grammatik  des  Schulrechnens,"  a  series 
of  text-books  which  cover  the  subject  from  elementary  arith- 
metic up  through  advanced  mathematics,  is  used  extensively 
throughout  the  high  schools.  It  is  especially  helpful  to  the 
teacher  in  presenting  the  subject,  and  contains  practical  and  con- 
crete problems  connected  with  the  daily  life  of  the  pupil.  Of  the 
texts  in  geometry  and  stereometry  Spiekersche's  "  Lehrbuch  der 
ebenen  Geometric"  (1908)  is  a  popular  book.  During  the  first 
two  years  of  the  high  school  Mahler's  text  is  used.  In  sterom- 
etry  the  texts  of  Kommerell,  Hauck,  Tubingen,  and  Bragg 
(1908)  are  studied.  In  trigonometry,  Burklen's  "Lehrbuch" 
(Heilbronn,  1897)  is  used.  Students ,  have  the  privilege  of 
selecting  the  text  to  be  used  for  the  study  of  spherical  trigonome- 
try. In  analytics  a  number  from  the  Sammlung  Goschen  is  used. 
Other  mathematics  is  studied  from  manuscripts. 

QUALIFICATION  OF  TEACHERS  OF  SECONDARY  SCHOOLS 

The  custom  of  giving  teachers'  examinations  was  inaugurated 
in  1846.  Examinations  were  given  in  German,  French,  nature- 
study,  literature,  geography,  and  mathematics,  the  last  named 
including  the  following  topics:  arithmetic,  algebra,  geometry, 
stereometry,  advanced  trigonometry,  and  practical  geometry,  e.g., 
the  use  of  measuring  instruments.  The  candidate  who  wished 


38  Teachers  College  Record  [104 

to  qualify  for  the,  Oberrealschule  had  to  pass  examinations  in 
spherical  trigonometry,  analytics,  calculus,  practical  geometry 
(advanced),  mechanics,  and  machine  construction  and  design. 
He  must  also  have  had  a  two  years'  course  in  some  polytechnic 
school  and  have  attended  a  university  for  one  year.  He  was 
required  to  take  an  oral  examination  in  the  presence  of  the  pro- 
fessor in  whose  charge  he  was  to  be  placed,  and  to  pass  an 
examination  on  the  teaching  of  mathematics.  The  mathematics 
teacher  was  required  to  pass  examinations,  not  only  in  the  sub- 
jects which  he  expected  to  teach,  but  in  other  subjects  as  well, 
as  he  was  expected  to  be  able  to  take  the  place  of  other  teachers 
in  case  of  absence.  Not  more  than  two  opportunities  to  take 
the  examinations  were  allowed  the  prospective  teacher. 

In  1907  a  new  system  of  examinations  affecting  the  third  from 
the  upper  class  of  the  Wurttemberg  schools  was  inaugurated. 
This  examination  was  required  of  everyone  and  covered  all  sub- 
jects then  being  pursued.  The  examinations  were  very  rigid  and 
consisted  of  both  oral  and  written  work.  It  was  made  possible 
for  one  who  passed  an  excellent  written  examination  to  be  ex- 
cused from  the  oral  one. 


CHAPTER  VI 
THE  SECONDARY  SCHOOLS  OF  BAVARIA1 

M.  J.  Leventhal 

The  object  of  this  report  by  Professor  Wieleitner  is  to  give  a 
survey  of  the  evolution  of  mathematics  in  the  Bavarian  high 
schools,  together  with  additional  information  concerning  the 
training  and  extension  work  of  teachers  in  Bavarian  Gymnasien, 
and  in  the  humanistic  schools  in  particular. 

The  work  done  in  the  latter  type  of  school  may  suggest  im- 
provement of  our  own  program.  The  school  year  in  Bavaria  is 
only  a  trifle  longer  than  ours,  extending  from  September  i8th 
to  July  I4th,  yet  the  work  done  is  far  superior  to  ours,  both  in 
character  and  content. 

With  regard  to  the  correlation  of  mathematics  with  other  allied 
sciences,  Bavaria  is  still  undecided.  Among  the  instructors, 
there  are  only  two  who  advocate  a  combination  of  all  the  natural 
science  departments.  Professor  Wieleitner  is  of  the  opinion  that 
the  teacher  should  teach  the  same  class  both  mathematics  and 
physics,  although  he  confesses  that  few  of  the  teachers  know 
mathematics  and  physics  equally  well.  ,  It  is  generally  recog- 
nized that  the  physicist  is  apt  to  treat  his  mathematics  too 
empirically,  and  the  mathematician  to  treat  his  too  mathemat- 
ically, but  nevertheless  it  is  the  general  feeling  that  this  is 
better  than  to  have  two  separate  capable  instructors  teach  the 
two  subjects  without  any  mutual  reference  to  each  other. 

At  present  there  are  two  scientific  groups  in  the  Bavarian 
schools:  (i)  mathematics  and  physics,  (2)  chemistry  and  biology. 
The  arranging  of  these  two  groups  is  considered  a  great  step 
forward.  For  a  long  period  in  Bavarian  Gymnasien  the  old- 


xDer  Mathematische  Unterricht  an  den  Hoheren  Lehranstalten  sowie 
Ausbildung  und  Fortbildung  der  Lehrkrafte  im  Konigreich  Bayern,  von 
Dr.  Heinrich  Wieleitner,  Leipzig  and  Berlin,  igro. 

105]  39 


4O  Teachers  College  Record  [106 

time  master,  trained  only  as  a  philologist,  taught  mathematics, 
physics,  French,  and  theology,  besides  his  Latin  and  Greek,  and 
the  establishing  of  these  groups  put  an  end  to  the  old  regime. 

THE  HUMANISTIC  SCHOOL 

This  type  of  school  deserves  our  special  attention,  for  it  is 
the  type  which  may  best  be  compared  to  our  average  high 
school.  In  the  colleges,  it  would  correspond  to  the  so-called 
"  arts  course."  From  the  Gymnasium,  jurists,  theologists,  phil- 
ologists, physicians,  and  officers  get  their  mathematical  knowl- 
edge, and  in  the  early  days  this  knowledge  was  very  slight. 

Up  to  1901  all  reforms  and  suggestions  that  were  proposed 
seemed  merely  to  cause  retrogression  in  the  school  program,  and 
to  foster  still  greater  conservatism  in  the  system.  In  the  year 
1901,  however,  a  definite  improvement  is  noticeable,  and  conse- 
quently we  shall  at  first  confine  our  attention  to  the  advance  that 
was  made  in  that  year.  We  shall  then  examine  the  new  sug- 
gestions and  reforms  proposed  by  Bavarian  educators  in  the 
attempt  to  ameliorate  the  status  of  mathematics  in  the  human- 
istic schools. 

The  classes  in  these  schools  are  numbered,  as  is  the  case  in  most 
German  high  schools.  For  our  purposes  we  shall  use  the  num- 
bers from  i  to  9  to  indicate  the  successive  school  years  as  usual, 
beginning  with  the  lowest  ( I ) .  For  the  sake  of  brevity,  we  shall 
not  outline  the  program  in  full,  but  shall  note  only  the  con- 
spicuous features.  It  will  be  our  task,  then,  to  determine 
whether  the  American  teachers  of  mathematics  can  gain  any- 
thing from  the  experience  of  the  schools  in  Bavaria. 

A  conspicuous  feature  of  the  program  of  the  Bavarian  schools, 
which  can  scarcely  escape  our  attention,  is  their  emphasis  upon 
mental  work.  The  Bavarian  student  can  work  problems  men- 
tally that  our  American  high-school  boys  would  find  hard  to 
solve  even  with  the  use  of  pencil  and  paper.  This  mental 
training  makes  the  former  more  acute,  and  makes  him  a  swifter 
and  more  accurate  calculator.  Besides,  despite  our  modern  psy- 
chologists, it  is  claimed  that  it  improves  his  power  of  memory. 
The  statement,  "  I'll  look  it  up,"  so  common  with  American  stu- 
dents is  not  in  vogue  in  Bavarian  humanistic  schools. 


107]     The  Present  Teaching  of  Mathematics  in  Germany      41 

Parentheses  are  taken  up  in  the  fourth  school  year.  When  we 
come  to  percentage  and  write  the  formula  A  =  B  (i  +  R),  our 
students  are  puzzled  to  know  the  significance  of  those  "  curved 
lines."  The  Bavarian  child  understands  them  as  soon  as  he 
takes  up  the  multiplication  of  integers.  In  algebra  not  much 
has  to  be  said  about  the  subject.  The  pupil  learns  very  early  to 
operate  with  x,  y,  and  a,  as  he  does  with  5,  10,  and  12. 

In  the  fifth  school  year,  common  and  decimal  fractions  are 
taken  up  simultaneously.  The  student  is  taught  to  see  clearly 
that  the  decimal  is  nothing  but  a  new  expression  for  the  ordinary 
fraction.  He  sees  the  identity  of  an  answer  when  given  in 
decimal  form,  in  the  operation  of  multiplication,  for  instance, 
with  the  answer,  when  all  the  elements  of  the  operation  are 
reduced  to  common  fractional  forms. 

Having  completed  the  work  in  percentage,  alligation,  and 
interest,  algebra  is  begun  in  the  eighth  school  year.  The  work 
is  such  as  we  have  in  the  last  year  of  some  of  our  best  ele- 
mentary schools.  At  this  time  they  also  begin  geometry.  It 
does  not  at  all  coincide  with  our  mensuration,  since  they  take  up 
congruence  of  triangles,  loci,  and  quadrilaterals  and  their  prop- 
erties. 

With  the  next  year,  we  find  the  beginning  of  what  corre- 
sponds to  our  high-school  course.  In  algebra  the  work  includes 
linear  equations,  such  proportion  as  is  needed  for  the  study 
of  the  similarity  of  triangles,  equations  involving  two  or  three 
unknowns,  theory  of  exponents,  extraction  of  the  square  root, 
and  quadratics  involving  one  variable.  In  geometry  we  find 
such  topics  as  the  following:  similarity,  relations  and  measure- 
ment of  rectilinear  figures,  equivalence,  computation  exercises, 
and  construction  problems. 

We  should  notice  in  this  arrangement  of  the  work  that  a 
program  bringing  out  a  perfect  correlation  of  algebra  and  geom- 
etry is  possible,  but  that  no  fusion  of  the  two  branches  in  one 
is  attempted.  Proportion  in  algebra  aids  the  student  in  under- 
standing similarity  in  geometry,  and  similar  figures  show  him  the 
practical  utility  of  the  "  Rule  of  three."  Furthermore,  a  few 
of  the  laws  of  algebra,  often  without  much  meaning  to  the  stu- 
dent, become  intelligible  and  full  of  significance  when  seen  from 


42  Teachers  College  Record  [108 

the  standpoint  of  geometry.  Correlation  without  destruction 
seems  the  dominant  idea. 

Logarithms,  postponed  in  our  course  to  the  last  high-school 
year,  are  taken  in  the  tenth  year.  Why  should  not  the  student 
be  able  to  use  this  handy  tool,  the  logarithmic  table,  to  work  out 
complicated  computations,  which  are  to  him  so  laborious?  The 
Bavarian  teachers  continue  their  plane  geometry  in  the  later 
years,  with  especial  emphasis  upon  algebraic-geometric  exercises, 
and  with  a  thorough  study  of  the  mensuration  of  plane  surfaces. 
As  soon  as  the  use  of  the  logarithmic  tables  is  understood,  the 
student's  mind  is  deemed  mature  enough  for  the  study  of 
trigonometry.  Important  goniometric  formulas  are  then  studied, 
and  the  measurement  of  solid  angles  and  crystals  is  particularly 
emphasized. 

Thus  the  numerical  work  prepares  the  student  for  computing 
the  lateral  areas  and  volumes  of  the  ordinary  solids  in  the  eleventh 
school  year.  In  general,  the  student  is  advanced  far  enough  to 
be  capable  of  the  appreciation  of  the  more  difficult  theorems  in 
solid  geometry,  such  as  Euler's  theorem,  and  the  prismatoid 
formula. 

It  is  thought  that  the  student  has  quite  enough  of  analytic  and 
synthetic  work  to  begin  his  analytic  geometry  in  the  secondary 
school.  At  present  this  subject  is  treated  in  various  schools 
according  to  a  set  program.  The  course  is  intended  merely  to 
give  the  pupil  a  training  in  the  fundamental  notions  and  methods 
of  analytics,  being  limited  to  a  study  of  the  co-ordinate  system 
and  its  applications  in  physics  and  statistics. 

It  would  be  interesting  to  mention  in  some  detail  certain  of  the 
topics  treated  in  the  course,  but  we  have  space  for  only  a  single 
illustration.  For  example,  the  following  equations  are  consid- 
ered, their  meaning  explained,  and  their  graphs  drawn  and  fully 
discussed  : 


y'  =  r',  (x—  a)'  +  (y—  b)»=r», 
x=rcosa,  y      =rsina, 

xl     y* 


and  y  =  sin  x. 


109]     The  Present  Teaching  of  Mathematics  in  Germany      43 

In  the  twelfth  school  year,  it  is  the  aim  to  collect  all  the  theory, 
show  the  connection  of  formulas,  and  demonstrate  their  use  in 
the  physical  sciences.  On  this  account  mathematical  geography 
is  taken  up,  together  with  Kepler's  laws,  Newton's  law  of  gravi- 
tation, and  the  use  of  co-ordinates  in  the  determination  of  the 
position  of  stars.  A  good  deal  of  mathematical  physics  is  intro- 
duced in  this  connection. 

Under  the  new  plan  proposed,  some  of  the  calculus  is  to  be 
introduced  in  the  eleventh  school  year,  and  in  the  following  year 
maxima  and  minima  and  a  study  of  differential  quotients  are  to 
be  studied. 

It  is  also  thought  that  it  would  not  be  too  radical  to  introduce 
trigonometry  in  the  ninth  or  tenth  year,  directly  after  the  study 
of  similarity  of  figures.  Many  relations  in  geometry  which  we 
express  to-day  by  proportion  could  still  better  be  expressed  by 
the  use  of  the  sine  and  the  cosine. 

SOLID  GEOMETRY 

It  has  been  proposed  to  put  some  of  the  simpler  work  in  vol- 
umes, points,  lines,  and  planes  a  little  earlier.  This,  however, 
depends  on  the  question  whether  it  would  not  be  more  profitable 
to  treat  the  plane  and  solid  geometries  together.  In  Italy,  since 
1900,  they  have  had  the  fusion  system  in  some  schools.  In 
Bavaria,  however,  they  have  no  book,  as  yet,  in  which  this 
plan  is  worked  out.  In  the  Italian  schools  they  have  Lazzeri 
and  Bassani's  "  Primi  Elementi  di  Geometria  "  and  one  or  two 
other  similar  works. 

These  reforms  have  been  brought  about  by  the  earnest  activity 
of  the  various  organizations  of  teachers.  All  plans  that  are  pro- 
posed are  investigated  by  the  Bavarian  section  of  the  Organiza- 
tion of  Teachers  for  the  Promotion  of  Mathematical  and  Physical 
Sciences.  Much  of  the  advance  is  also  due  to  an  active  society 
in  Munich,  the  Munich  Society  of  Teachers.  Since  1907,  a 
number  of  similar  associations  have  been  formed,  which,  besides 
investigating  various  reforms,  have  arranged  for  numerous 
scientific  reports  and  lectures.  The  spirit  of  the  Bavarian  teach- 
ers is  admirable,  and  their  achievements  are  worthy  of  emulation 
in  our  country. 


CHAPTER  VII 
THE  HIGHER  SCHOOLS  FOR  BOYS  IN  PRUSSIA1 

Robert    King    Atwell 

The  well-known  American  authority,  Professor  J.  W.  A. 
Young,  who  is  quoted  by  Dr.  Lietzmann  in  this  report,  explains 
the  superiority  of  the  German  schools  over  the  American  schools 
in  this  manner :  "  The  causes  of  the  excellence  of  the  Prussian 
work  in  mathematics  may  be  classed  under  three  heads :  ( i )  The 
central  legislation  and  supervision.  (2)  The  preparation  and 
status  of  the  teachers.  (3)  The  method  of  instruction." '  Of 
these  three  contributing  causes,  the  first  may  well  deserve  a 
little  attention  at  this  time,  since  it  is  not  discussed  in  any  other 
chapter. 

EDUCATIONAL  ADMINISTRATION 

The  educational  work  of  the  state  is  under  the  direction  of  the 
Ministry  of  Spiritual,  Educational  and  Medicinal  Affairs,  estab- 
lished in  1817.  Besides  the  Minister,  there  are  an  under-State 
Secretary  and  Division  Directors.  In  addition  to  these  officials, 
each  division  includes  reporting  boards  and  assistants  selected 
from  the  body  of  professional  men  and  administrators.  The  Min- 
ister reaches  his  decisions  independently,  since  "  every  ministerial 
decision  is  considered  as  coming  from  the  Minister  himself."  He 
is  answerable  to  the  King  and  the  Diet.  Since  Prussia  has  no 
educational  statutes,  the  policy  of  higher  public  instruction  is 
determined  by  ministerial  decrees. 

The  individual  schools  are  not  directly  answerable  to  the 
ministry.  There  are  intermediate  authorities,  namely,  the  Pro- 
vincial School  Boards.  There  are  twelve  of  these  bodies  in  the 
capital  of  the  Prussian  provinces.  The  Presidents  are  the  Gen- 
eral Presidents  of  the  province.  The  Directors  (or  vice-presi- 

1  Die   Organisation   des   Mathematischen   Unterrichts  an   den   Hoheren 

Knabenschulen  in  Preussen.  von  W.  Lietzmann.  Leipzig  und  Berlin,  1910. 

*  J.  W.  A.  Young :    The  Teaching  of  Mathematics  in  Prussia.    N.  Y.  1900. 

44  [no 


in]     The  Present  Teaching  of  Mathematics  in  Germany      45 

dents)  are  in  some  cases  schoolmen,  but  usually  administrative 
officers.  The  number  of  members  of  the  Provincial  School 
Boards  varies  from  one  to  five,  if  we  include  only  the  "  Dezer- 
nenten  "l  'for  the  higher  public  instruction ;  to  this  number  are 
sometimes  added  pedagogical  experts. 

The  Provincial  School  Boards  become  acquainted  with  the 
schools  of  their  districts  through  short  visits  as  well  as  by  con- 
ducting the  final  examinations  of  the  "  higher  secondary  "  insti- 
tutions, and  by  granting  the  certificate  of  the  Lower  Secondary 
School  which  indicates  that  the  candidate's  final  examination 
has  been  passed,  and  entitles  him  to  a  diminution  of  one  year 
of  military  service.  To  each  member  of  the  Board  a  certain 
number  of  students  of  his  district  is  assigned,  to  be  under  his 
special  charge.  He  watches  these  students  during  their  entire 
course.  These  lists  are  revised  every  three  or  four  years. 

All  important  matters,  such  as  the  maintenance  of  instruction, 
the  selection  of  the  subject  matter  of  the  readings  in  the  lan- 
guages, the  introduction  of  text-books,  and  the  like,  are  subject 
to  the  ratification  of  the  Provincial  School  Boards.  They  also 
appoint  the  teachers  in  the  royal  institutions  and  confirm  them 
in  the  city  institutions. 

There  are  numerous  non-state  schools,  mainly  city  high 
schools.  The  private  or  the  religious  schools  of  Prussia  are 
not  considered  in  this  report.  In  the  west,  these  schools  for  the 
most  part  have  "  guardians  "  or  trustees ;  in  the  east,  the  magis- 
trate exercises  his  rule  directly;  in  the  large  cities,  a  technical 
assistant  is  assigned  to  the  city  high  schools.  The  guardians  and 
magistrates  exercise  the  power  of  selecting  the  directors  and 
teachers,  as  well  as  of  preparing  the  budget.  The  budget  is 
afterward  referred  to  the  board  appointed  by  the  city. 

The  administration  of  the  various  schools  is  in  the  hands  of 
the  directors  who  are  not  only  administrative  officers  but  teach- 
ers as  well,  required  to  teach  a  certain  number  of  hours. 

Concerning  the  relation  of  the  director  to  his  teachers  no 
universal  rules  have  been  laid  down,  though  some  such  regula- 
tions are  now  being  considered.  In  fact,  in  all  the  provinces 
there  is  some  "  service  instruction,"  by  which  it  is  very  often 

1  Appointees. 


46  Teachers  College  Record  [112 

stated  that  the  director  is  omnipotent  and  the  teacher  has  no 
rights;  but  these  strict  decrees  frequently  exist  only  on  paper. 
The  relations  between  the  director  and  the  teaching  staff  are 
largely  determined  by  the  personality  of  those  concerned,  and 
not  by  official  decrees. 

Every  four  years,  in  all  the  provinces  of  Prussia  except 
Brandenburg  (as  well  as  Hessen-Nassau  until  very  recently),  the 
directors  of  the  higher  boya'  schools  meet  in  the  General  Assem- 
bly of  Directors,  which  is  attended  also  by  the  members  of  the 
provincial  school  board  and  councillors  from  the  ministry. 

THE  REFORM  MOVEMENT 

As  early  as  1899  tne  seventh  Directors'  Assembly  of  the 
Rhine  sought  expert  advice  upon  the  question:  What  sugges- 
tions for  the  bettering  of  mathematical  instruction,  recently  pub- 
lished, deserve  to  be  put  in  practice  in  the  higher  schools  ? 

The  reform  movement  in  mathematical  instruction  finds  its 
most  definite  expression  in  the  so-called  Meran  and  Stuttgart 
"  Proposals  "  of  the  Commission  on  Instruction,  appointed  by 
the  Association  of  German  Natural  Philosophers  and  Physicians. 
The  first  conference  of  this  body  was  held  in  1900.  After  that 
it  met  yearly,  but  the  first  meeting  considered  worthy  of  any 
extended  report  was  held  in  1908. 

The  year  1908  is  noteworthy  for  the  appointment  of  the  Ger- 
man Commission  for  Mathematics  and  Science  Instruction.  The 
commission  included  among  its  members  such  prominent  mathe- 
maticians as  Professors  Gutzmer,  Klein,  Poske,  Schotten, 
Stackel,  Thaer,  and  Treutlein. 

Of  the  questions  discussed  by  this  commission  the  following 
are  especially  worthy  of  note: 

1.  A   report  concerning  the  mathematical  instruction   in  the 
"  higher  institutions  of  more  classes."     Meran,  1905. 

2.  The  mathematical  and  natural  science  instruction  in  Re- 
formschulen.     Stuttgart,  1906. 

3.  The  mathematical  and  natural  science  instruction  in  the  six- 
class  Realstalten.     Stuttgart,  1906. 

The  suggestions  of  the  Commission  on  Instruction  embody  two 
phases  of  the  mathematical  instruction  in  the  higher  schools : 


113]     The  Present  Teaching  of  Mathematics  in  Germany      47 

1.  The  strengthening  of  the  power  of  spatial  conception. 

2.  Training  in  the  habit  of  thinking  in  functions. 

In  their  various  ways  the  members  of  the  Commission  seek 
to  emphasize  the  following  ideals : 

1.  To  adapt  the  course  more  than  formerly  to  the  natural 
steps  of  mental  development.    This  psychological  principle  mani- 
fests  itself   in   the   emphasis   upon   propaedeutic   instruction  in 
arithmetic  and  geometry,  in  the  demand  for  a  gradual  transition 
from  intuitive  to  deductive  treatment. 

2.  "  To  bring  to  the  attention  of  teachers  of  mathematics  the 
possible  solution  of  the  problems  of  the  visible  world  around 
us."    This  utilitarian  principle  shows  its  influence  in  the  search 
for  applications  of  mathematics. 

3.  To  make  the  subject  matter  of  the  course  within  itself, 
from  class  to  class,  more  and  more  coherent.     This   didactic 
principle  leads  to  the  concentration  of  the  instruction  in  the  ag- 
gregate around  one  fundamental  idea,  the  function,  in  the  sub- 
jects of  algebra  and  geometry. 

For  the  purpose  of  showing  the  influence  of  the  reform,  the 
various  secondary  institutions  may  be  divided  into  groups,  the 
classification  being  made  according  to  their  attitude  toward  the 
adoption  of  the  suggestion.  For  example,  the  classification  of 
the  upper  classes  (oberstufe)  of  these  institutions  is  as  follows: 

I.  Those  schools  which  refused  to  introduce  the  idea  of  func- 
tion. 

II.  Those  which  believed  in  a  late  (uncertain)  introduction  of 
the  function  concept. 

a.  Without  calculus. 

b.  With  differential  calculus. 

c.  With  differential  and  integral  calculus. 

III.  Those  which  preferred  a  gradual  (regular)  introduction  of 
the  function  concept. 

a.  Without  calculus. 

b.  Wtih  differential  calculus  in  the  Prima. 

c.  With  differential  and  integral  calculus  in  the  Prima. 

d.  With  integral   calculus  already  in  the  upper  half 
of  the  class  below  the  Prima. 


48  Teachers  College  Record  [114 

In  making  up  this  list,  the  programs  of  over  six  hundred  insti- 
tutions were  examined.  There  were  but  few  institutions  which 
were  radically  "  reformed,"  while  the  remaining  institutions  were 
about  equally  divided  between  those  that  adopted  the  reform 
to  a  moderate  extent,  and  those  that  ignored  the  reform  alto- 
gether. 

The  question  as  to  how  far  this  most  recent  reform  agitation 
has  penetrated  into  the  schools  can  be  estimated  roughly  from 
another  grouping  of  institutions  which  includes  only  those  which 
show  changes  in  the  programs. 

On  the  basis  of  the  Easter  program  of  1909,  a  list  has  been 
arranged  which  gives  merely  an  approximation,  since  only  the 
programs  dispatched  in  August,  1909,  could  be  considered.  Ac- 
cording to  these  data,  the  introduction  of  the  idea  of  function 
was  noted  in  twenty-four  Gymnasien,  thirty-five  Realgymnasien, 
thirty-four  Oberrealschulen,  and  five  Realschulen,  making  a  total 
of  ninety-eight  institutions. 

The  recent  additions  to  the  requirements  in  mathematics  are 
nothing  new  in  themselves.  During  the  last  twenty  or  thirty 
years,  certain  far-sighted  men  have  already  done  what  the  Meran 
Proposal  calls  for.  Proof  of  this  will  be  found  in  numerous 
instances  by  a  later  historian  of  the  Reform  Movement,  who, 
being  removed  by  a  greater  interval  of  time  from  these  move- 
ments, which  are  at  present  in  such  a  state  of  transition,  will  be 
able  to  make  a  completely  objective  judgment. 

The  psychological  principle  referred  to  above  has  had  zealous 
advocates  before  the  present  agitation  for  reform,  to  name  only 
Treutlein  in  Baden  and  Hofler  in  Austria.  There  are  many 
mathematicians,  however,  who  do  not  agree  on  this  point,  al- 
though otherwise  heartily  in  sympathy  with  the  reform  move- 
ment. 

The  suggestions  embodied  in  these  "  Proposals "  have  been 
made  before.  Among  the  instances  which  might  be  cited  is  that 
apparently  almost  forgotten  report  of  A.  von  Oettingen  concern- 
ing the  mathematical  instruction  in  the  schools  of  Dorpat.  In  it 
we  find  the  following  requirements:1  (i)  The  introduction  of 


1  Report    on    the    anniversary    of   the    founding    of    the    University   of 
Dorpat,   1873. 


115]     The  Present  Teaching  of  Mathematics  in  Germany      49 

the  relations  of  variable  quantities ;  in  short,  the  idea  of  function. 
(2)  The  bare  elements  of  analytic  geometry  as  far  as  the  calculus. 
However,  in  the  opinion  of  Dr.  Lietzmann,  it  is  entirely  wrong 
to  assert  that  the  proposed  reforms  in  the  instruction  in  mathe- 
matics teach  nothing  new,  that  what  is  now  desired  was  long 
ago  brought  out  by  others.  Hofler  once  pointed  out  to  some  of 
the  men  of  greatest  influence  in  mathematical  circles  that  the 
highest  compliment  that  can  be  paid  to  the  reform  movement 
is  that  it  contains  no  items  which  are  fundamentally  new,  and 
that  no  matter  how  many  such  forerunners  may  be  found,  it 
is  the  harmonizing  of  all  these  proposals,  which  formerly  were 
often  sharply  opposed,  into  one  powerful  impulse,  as  well  as 
the  co-operation  of  so  many  forceful  personalities,  which  makes 
this  movement  one  for  which  no  analogy  can  be  found  in  the 
history  of  mathematics. 


CHAPTER  VIII 

THE  SECONDARY  SCHOOLS  OF  ELSASS  AND 
LOTHRINGEN1 

Maurice   Levine 

In  the  first  chapter  Dr.  Wirz  describes  the  present  organiza- 
tion of  the  higher  schools  in  Elsass-Lothringen  (Alsace-Lor- 
raine). This  is  followed  by  a  historical  and  critical  survey  of 
the  development  of  instruction  in  mathematics,  especially  with 
regard  to  the  curriculum,  from  the  time  of  the  French  control 
in  1870  to  the  present  day.  The  third  chapter  deals  with  the 
methods  of  instruction  employed  at  the  present  time.  The  part 
that  text-books  play,  the  question  of  propaedeutic  instruction,  the 
use  of  models,  and  practical  exercises,  are  all  discussed.  The 
fourth  chapter  is  devoted  to  a  discussion  of  the  reform  move- 
ment in  Elsass-Lothringen.  The  author  gives  the  opinions  of 
various  colleagues  as  to  the  use  and  scope  of  the  function-con- 
cept, the  introduction  of  the  differential  and  integral  calculus  in 
the  secondary  school,  the  cutting  down  of  the  formal  operations, 
the  simultaneous  treatment  of  planimetry  and  stereometry, 
geometrical  drawing,  the  use  of  historical  material,  the  instruc- 
tion in  the  upper  classes,  and  the  final  examinations.  The  last 
chapter  discusses  the  preparation  of  teachers  of  mathematics  in 
the  higher  schools. 

ORGANIZATION 

The  higher  schools  of  Elsass-Lothringen  are  under  the  super- 
vision of  a  central  board,  the  head  of  which  holds  a  separate 
seat  in  the  ministry.  With  him  are  associated  three  assistants 
(Oberschulrate).  The  board  appoints  the  "  Direktor  "  or  presi- 
dent of  each  school,  a  special  commission  of  the  board  appoints 

'Der  Mathematische  Unterricht  an  den  Hoheren  Knabenschulen  sowie 
die  Ausbildung  der  Lehramtskandidaten  in  Elsass-Lothringen,  von  Pro- 
fessor J.  Wirz,  Direktor  der  Oberrealschule  in  Colmar,  Leipzig,  1911. 

50 


1 1/]     The  Present  Teaching  of  Mathematics  in  Germany      51 

the  instructors,  and  another  committee  has  charge  of  the  "  Reife- 
priifung."  The  schools  of  gymnasial  character  are  organized 
according  to  a  definite  plan.  The  text-books  are  all  prescribed, 
and  the  courses  are  limited  in  a  general  way  both  as  to  content 
and  method. 

There  are  twenty-eight  state  schools  (14  Gymnasien,  i  Pro- 
gymnasium,  6  Oberrealschulen,  7  Realschulen)  and  9  private 
schools  (6  of  which  are  under  religious  control).  In  Novem- 
ber, 1910,  the  registry  of  these  schools  numbered  10,700  stu- 
dents (including  70  girls)  who  were  distributed  as  follows: 
5,186  at  the  Gymnasien,  301  at  the  Gymnasien  with  "Real" 
courses,  and  5,213  at  the  Realschulen  and  Oberrealschulen. 

THE  SCHOOL  SYSTEM  UNDER  FRENCH  CONTROL 

There  were  no  separate  schools  for  humanistic  and  realistic 
courses  prior  to  the  loss  of  the  country  by  France  (1870),  but 
the  schools  had  a  twofold  object.  The  humanistic  course  was 
nine  years  in  length,  and  the  classes  were  called  huitieme,  sep- 
ticmc,  sixicme,  cinquieme,  quatrieme,  troisieme,  seconde  rhetorique, 
and  philosophic.  This  course  prepared  the  student  for  the  bac- 
calaureat  es-lettres.  After  the  classe  seconde,  the  course  was 
divided  into  two  parts :  the  language-history  group  with  classes 
rhetorique  and  philosophic,  and  the  mathematics-science  group 
with  the  two  classes  de  mathematiques  elementaires,  to  which  was 
added  in  a  few  schools  the  classe  de  mathematiques  speciales.  The 
second  group  prepared  the  student  for  entrance  to  the  scientific, 
naval  and  polytechnic  schools. 

The  realistic  course,  preparing  for  the  practical  vocations,  was 
a  three  years'  course  and  started  in  the  quatrieme.  The  curricula 
varied  in  the  different  towns  according  to  the  industries  that 
were  locally  the  most  important.  The  courses  for  the  two 
groups  of  schools  were  as  follows: 

T.  HUMANISTIC  GROUP.  Until  the  quatrieme:  ordinary  arith- 
metic. 

Quatrieme:  some  simple  geometry. 

Troisieme  (two  periods,  i.e.,  of  two  consecutive  hours  each)  : 
factoring,  prime  numbers,  partnership,  G.C.D.,  fractions, 
decimal  fractions,  proportion,  discount,  business  arithmetic,  lit- 


52  Teachers  College  Record  [118 

eral  counting  (introduction).  In  geometry:  theorems  on  tri- 
angles, quadrilaterals,  circles,  similar  triangles,  proportion,  sur- 
face, fundamental  constructions. 

Scconde  (two  periods1)  :  algebraic  operations,  equations,  rela- 
tions between  algebraic  and  planimetric  exercises.  In  geometry : 
regular  polygons,  circles,  plane  figures,  limits,  areas  of  similar 
figures,  field  measurements.  In  stereometry:  plane  and  line, 
polyhedrons,  prisms,  pyramids,  areas  of  parallel  surfaces, 
sections. 

Rhctorique  (one  period):  In  geometry:  cylinders,  right  cone, 
sphere,  mathematical  geography,  maps  (various  projection  sys- 
tems). 

Philosophic  (three  periods,  1st  semester,  two  periods,  2nd 
semester).  No  definite  plan  was  prescribed.  The  work  in  this 
class  was  left  to  the  instructor,  but  it  was  usual  to  review  the 
work  of  the  former  classes,  and  to  give  new  work  in  logarithms, 
with  the  use  of  tables,  and  also  to  take  up  the  study  of  similar 
figures  in  space. 

The  work  in  the  clause  troisieme  was  altogether  too  extensive, 
and  was  really  a  paper  course,  the  class  not  being  able  to  ac- 
complish even  half  of  it  satisfactorily.  There  was  very  little 
algebra  throughout,  and  no  trigonometry  at  all. 

Classes  de  mathematiques  elcmen-taires  (five  periods)  :  Arith- 
metic (a  term  covering  algebraic  work  with  numbers)  :  review 
and  further  development ;  imaginary  quantities,  maxima  and 
minima  for  quadratics,  progressions,  theory  of  logarithms. 
Geometry :  review ;  inscribed  figures,  original  problems.  Stere- 
ometry (practically  the  same  as  in  the  Prussian  Gymnasium)  : 
spherical  triangles,  ellipse  (fundamental  properties),  definition 
of  tangent  to  a  curve,  tangent  to  conies,  normals.  Plane  trigo- 
nometry :  trigonometric  lines  and  functions,  tables,  triangles,  dis- 
tances and  angles  of  inaccessible  points.  Descriptive  geometry 
and  mechanics:  only  the  elements  of  the  subjects. 

Classe  de  mathematiques  spcciales  (six  periods)  :  Arithmetic 
and  algebra:  review  and  further  development;  higher  equations, 
irrational  numbers,  incommensurability,  series,  combinations, 
binomial  theorem,  logarithms,  exponential  equations,  operations 
on  circular  functions,  development  of  log  (i  +  x)  and  arc  tan  x 

1  A  period  or  classe  means  two  consecutive  hours  of  work. 


II9J     The  Present  Teaching  of  Mathematics  in  Germany      53 

« 

in  series,  theory  of  equations,  interpolation  formulas.  Geometry : 
review  and  further  development  of  planimetry  and  stereometry ; 
conic  sections,  spherical  triangles,  polar  triangles,  congruence  by 
symmetry.  Trigonometry:  review  and  further  development; 
trigonometric  solutions  of  equations  of  the  second  and  third 
degrees,  spherical  trigonometry,  applications  to  geodesy,  prac- 
tical surveying.  Plane  analytic  geometry,  with  the  elements  of 
solid  geometry.  Descriptive  geometry. 

2.  REALISTIC  GROUP  :  First  year :  arithmetic,  plane  geometry, 
work  outdoors,  line  drawing,  commercial  arithmetic,  introduction 
to  bookkeeping.  Second  year :  commercial  arithmetic,  bookkeep- 
ing, beginning  algebra  and  solid  geometry,  field  work.  Third 
year:  completion  of  elementary  algebra,  bookkeeping,  finance, 
accounting,  trigonometry,  the  ordinary  curves,  descriptive 
geometry. 

Both  the  mathematics-science  and  realistic  programs  were  very 
thorough  on  paper,  but  the  schools  were  all  in  a  confused  state 
when  Germany  assumed  control  in  1871. 

GERMAN  CONTROL  FROM  1871  TO  THE  PRESENT  TIME 
The  control  of  the  schools  was  immediately  taken  out  of  the 
hands  of  the  academies  and  inspectors,  and  put  in  charge  of 
the  governor-general.  Higher  school  instruction  was  defined, 
but  it  took  about  a  year  for  the  program  to  be  understood.  Dr. 
Baummeister,  former  Gymnasium  director  at  Strassburg,  was 
summoned  to  build  up  the  system.  He  completely  revolutionized 
it,  imported  German  teachers  as  directors  and  instructors,  and 
made  the  following  changes  in  spite  of  the  opposition  and  preju- 
dices of  the  people,  including  the  older  teachers  and  the  pupils. 

(1)  Introduction  (from  Quinta  down)  of  the  natural  sciences, 
modern  languages,  physics,  and  chemistry.    The  old  mathematics 
program  was  allowed  to  remain  unchanged  for  a  time. 

(2)  Creation  of  the  Landesschulkonferenz. 

(3)  Establishment  of  a  definite  program. 

(a)  Latin  became  obligatory  throughout   the  course  of  the 
Gymnasium. 

(b)  The  humanistic  and  realistic  groups  were  made  Prussian 
instead  of  French.    The  two  groups  divided  in  the  Quarta.    The 
humanistic  course,  the  Gymnasium,  put  emphasis  on  Latin  and 


54 


Teachers  College  Record 


[120 


Greek,  while  the  realistic  course,  the  Realgymnasium,  empha- 
sized mathematics  and  the  sciences. 


VI 

r±. 

V 

Gymnasium — Latin  and  Greek 


Lat.  and  French. 

Realgymnasium — Latin  decreased,  mathematics,  English  and 
natural  science  increased. 

In  1872,  the  Abiturientenexamen  were  introduced  and  were 
almost  exactly  like  those  in  Prussia.  The  reform  in  this  year 
required  the  students  in  the  Gymnasium  to  hand  in  problems 
in  arithmetic  and  algebra  fortnightly,  while  those  in  the  Real- 
gymnasium  had  to  hand  them  in  weekly. 

Between  1873  and  1878,  many  important  reforms  were  intro- 
duced. All  primary  and  secondary  schools  were  placed  under 
the  absolute  control  of  the  state.  The  three  schools,  Gymnasium 
(9  years),  Realgymnasium  (9  years),  and  Realschule  (7  years) 
were  defined,  but  all  were  for  the  time  being  classed  as  Gym- 
nasien.  The  number  of  hours  per  week  allowed  to  mathematics 
were  as  follows: 

CLASS    No.  OF'Houas 
Gymnasium  All  3-4 

f  VI-V  3-i 

Realgymnasium  \    VI-III  6 

II-I  5 

VI-V  4 


Realachule 


IV-I 


0 


The  examination  in  mathematics  called  for  the  solutions  of  prob- 
lems in  all  the  different  fields  studied. 

In  1882,  Governor  Manteuffel  called  a  commission  of  educa- 
tors and  medical  experts  to  consider  whether  or  not  too  much 
work  was  being  demanded  of  the  pupils.  As  a  result  of  this 
commission's  work,  the  following  changes  were  made : 

( i )  The  higher  schools  were  grouped  as  follows : 

(a)  Gymnasium   (9  years),  Progymnasium  (6  years) 

and  Lateinschule  (3  years). 

(b)  Realschule. 


i2i  ]     The  Present  Teaching  of  Mathematics  in  Germany      55 

(2)  The  number  of  hours  of  instruction  was  cut  down  to  the 

following : 

In  VI  and  V,  27-28  hours  weekly. 

In  IV  and  III,  30  hours  weekly. 

In  the  other  classes,  32-34  hours  weekly. 

(3)  The  number  of  hours  of  home  work  was  limited: 

For  VI  and  V,  8  hours  weekly. 
For  IV  and  III,  12  hours  weekly. 
For  II  and  I,  12-18  hours  weekly. 

(4)  The   program   was   shortened    considerably.     In   mathe- 
matics, the  Prima  students  could  elect  spherical  trigonometry  and 
analytic  and  descriptive  geometry. 

Until  1898,  the  number  of  hours  in  mathematics  was  grad- 
ually increased  with  each  new  reform. 

In  1898,  the  mathematics  to  be  taught  in  each  class  of  the 
Gymnasium,  Realgymnasium,  and  Realschule  was  sharply  de- 
fined, and  the  program  is  now  in  use,  except  for  a  few  changes. 
It  is  given  below  in  full. 

PROGRAM  OF  THE  GYMNASIUM  AND  REALGYMNASIUM 

SEXTA.  Four  fundamental  operations  with  known  and  un- 
known integers ;  introduction  to  the  German  tables  (the  metric 
system)  ;  time  reckoning;  reduction. 

QUINTA.  Common  fractions ;  short  division  ;  long  division ; 
operations  with  common  fractions ;  reduction  of  fractions. 

QUARTA.  Arithmetic:  decimal  fractions,  percentage,  and  prac- 
tical examples.  Geometry:  the  elements  of  planimetry  ending 
with  the  fourth  congruence  theorem ;  elementary  exercises. 

UNTERTERTIA.  Algebra :  the  four  fundamental  operations  with 
algebraic  magnitudes ;  simple  equations  in  one  unknown ;  exer- 
cises. Geometry :  the  parallelogram,  circle,  construction  exercises. 

OBERTERTIA.  Algebra :  proportion ;  powers  with  positive  inte- 
gers ;  simple  equation  in  one  and  two  unknowns.  Geometry : 
surfaces ;  transformations ;  areas  of  right-angled  figures ;  pro- 
portionality of  lines ;  construction  exercises. 

UNTERSECUNDA.  Algebra :  involution  and  evolution ;  square 
root ;  simple  equations  in  several  unknowns ;  quadratic  equations 
in  one  unknown.  Geometry:  similarity:  measurement  of  the 


56  Teachers  College  Record  [122 

circle;  construction  exercises,  also  exercises  from  the  field  of 
algebraic  geometry. 

OBERSECUNDA.  Algebra:  quadratic  equations  with  one  un- 
known ;  easy  equations  with  two  unknowns ;  logarithms ;  arith- 
metic and  geometric  progressions  with  applications  to  interest  and 
annuities.  Geometry:  trigonometry,  right-angled  triangle,  sine 
and  cosine  theorems ;  harmonic  points  and  rays ;  similarity. 

UNTERPRIMA.  Algebra:  difficult  exercises  in  simultaneous 
quadratics ;  simple  exercises  in  maxima  and  minima.  Geometry : 
goniometry,  application  to  solution  of  triangle;  stereometry; 
planimetric  exercises. 

OBERPRIMA.  Algebra:  combinations;  probability  and  chance; 
binomial  theorem.  Geometry :  mathematical  geography  including 
necessary  theorems  of  spherical  trigonometry ;  fundamental  prop- 
erties of  the  ellipse,  parabola  and  hyperbola;  exercises. 

PROGRAM  OF  THE  REALSCHULE  AND  OBERREALSCHULE 

6.  REALKLASSE.    Same  as  Sexta  in  Gymnasium. 

5.  REALKLASSE.  Same  as  Quinta  in  Gymnasium,  and  also  the 
fundamental  operations  with  decimal  fractions. 

4.  REALKLASSE.  Arithmetic  (4  hours  in  the  ist  semester,  3 
hours  in  the  2nd  semester)  :  decimals,  changing  decimals  to  com- 
mon fractions ;  percentage,  interest,  partnership  and  mixtures. 
Geometry  (2  hours  in  the  ist  semester,  3  hours  in  the  2nd  sem- 
ester) same  as  Quarta  in  the  Gymnasium. 

3.  REALKLASSE.  Arithmetic  (i  hour)  :  commercial  arithmetic, 
weights  and  measures.  Algebra  (2  hours)  :  four  fundamental 
operations  with  algebraic  magnitudes ;  simple  equations  in  one 
unknown.  Geometry  (2  hours)  :  parallelogram;  circle;  construc- 
tions. 

2.  REALKLASSE.  Algebra  (3  hours)  :  proportion;  powers  and 
real  roots;  square  root;  simple  simultaneous  equations  in  two 
unknowns.  Geometry  (2  hours)  :  surfaces,  transformations,  and 
computations;  proportion  (introduction  to  similar  figures) ;  con- 
structions. 

i.  REALKLASSE.  Algebra  (2  hours)  :  imaginary  roots,  loga- 
rithms ;  quadratics  in  one  unknown ;  exponential  equations. 
Geometry  (3  hours):  planimetry,  similarity,  circle;  construe- 


123]     The  Present  Teaching  of  Mathematics  in  Germany      57 

tions ;  algebraic  geometry.  Trigonometry :  right-angled  triangle, 
sine  and  cosine  theorems.  Stereometry :  simple  solids. 

3.  OBERREALKLASSE.  Algebra :  quadratic  equations  in  two  un- 
knowns ;  arithmetic  and  geometric  progressions  with  applications 
to  compound  interest  and  annuities.  Geometry :  trigonometry, 
goniometry ;  simple  goniometric  equations  ;  solutions  of  triangles ; 
geodetic  exercises ;  planimetry,  harmonic  points  and  rays ;  simi- 
larity; stereometry. 

2.  OBERREALKLASSE.  Algebra :  exercises  in  maxima  and  min- 
ima; arithmetic  series  of  higher  order;  graphic  numbers;  com- 
binations ;  probability  and  error ;  higher  equations  in  the  quad- 
ratic form;  cubics.  Geometry:  spherical  trigonometry  with  ap- 
plications to  mathematical  geography  and  geodesy;  analytic 
geometry  of  the  point,  straight  line,  and  circle. 

i.  OBERREALKLASSE.  Algebra:  the  binomial  theorem,  De 
Moivre's  theorem;  infinite  series  with  applications  to  geodesy; 
numerical  equations  of  higher  degree.  Geometry :  analytic  geom- 
etry of  the  parabola,  ellipse,  hyperbola;  projective  geometry  of 
the  conic  sections. 

This  Lehrplan  of  1898  was  similar  to  the  Prussian  one.  The 
Gymnasium  course  included  analytics  and  spherical  trigonometry 
with  an  elective  in  descriptive  geometry,  graphical  statics,  and 
geodesy.  Projective  and  descriptive  geometry  were  required  in 
the  Oberrealschule.  Although  the  new  changes  were  in  line  with 
the  reform  movement,  there  was  no  calculus  at  all. 

The  last  great  changes  were  made  in  1905.  The  higher  schools 
were  divided  into  two  groups : 

I.  Gymnasium  (9  years),  Progymnasium  (6  years). 

II.  Realgymnasium  (9  years),  Oberrealschule  (9  years),  Real- 
schule  (6  years). 

The  mathematics  for  both  the  Gymnasium  and  Realgym- 
nasium was  the  same  (35  hours).  The  Oberrealschule  did 
much  more  in  this  field  (45  hours).  A  few  changes  in  the 
curriculum  may  be  mentioned.  Plane  and  spherical  trigonom- 
etry, and  plane  analytic  geometry  were  made  complete,  and  choice 
was  given  in  the  Realgymnasium  between  descriptive  geometry 
and  freehand  drawing.  The  differential  and  integral  calculus  is 
still  lacking,  but  projective  geometry  takes  its  place.  The  reform 


58  Teachers  College  Record  [124 

movement  makes  itself  felt  in  the  study  of  particular  parts  of 
the  calculus,  such  as  maxima  and  minima,  and  series. 

The  Reifepriifung  was  made  very  much  like  the  Prussian, 
except  that  no  one  might  be  completely  exempted  from  the  oral 
examination.  The  written  examination  in  the  three  schools  ex- 
tended over  five  or  six  hours,  and  consisted  of: 

1.  A  German  composition  on  a  theme  within  the  mental  power 
of  each  student; 

2.  Four  problems  from  the  different  parts  of  mathematics. 
To  this  was  added: 

(a)  For  the  Gymnasium:  translation  from  German  to  Latin 
and  from  Greek  to  German. 

(b)  For  the  Realgymnasium :  a  French  composition  or  a  trans- 
lation from  Latin  to  German. 

(c)  For  the  Oberrealschule :  a  French  composition  and  a  trans- 
lation from  German  to  English. 

It  is  the  aim  in  teaching  mathematics  in  the  Gymnasium  and 
Realgymnasium  to  enable  the  student  to  know  his  algebra  up  to 
the  binomial  theorem  and  quadratics,  his  plane  geometry,  plane 
trigonometry,  and  solid  geometry,  and  to  apply  this  knowledge 
to  the  solution  of  simple  problems.  In  the  Oberrealschule,  the 
requirements  are  the  development  of  the  most  important  series, 
solution  of  the  cubic  equation,  plane  and  solid  geometry,  plane 
and  spherical  trigonometry,  analytic  and  projective  geometry  of 
the  plane,  and  applications  to  solutions  of  problems.  In  this 
respect,  the  Gymnasium  course  is  considerably  above  our  ordin- 
ary high  school  course  in  mathematics,  although  in  a  good  Amer- 
ican high  school  the  student  may  elect  a  little  more  advanced 
algebra.  The  Oberrealschule  goes  as  far  in  mathematics  as 
the  freshman  class  in  a  good  engineering  school  in  this  country, 
except  that  in  some  schools  the  freshman  completes  his  differ- 
ential and  integral  calculus  at  the  end  of  the  first  year. 

METHODS  OF  INSTRUCTION 

There  are  no  definite  text-books  used,  but  new  books  cannot 
be  placed  on  the  official  ligf  without  the  approval  of  the  Royal 
Minister.  In  geometry,  the  text-book  is  used  for  extension  and 
review,  and  for  a  survey  of  the  whole  field.  The  instructor  does 


125]     The  Present  Teaching  of  Mathematics  in  Germany      59 

not  follow  the  book,  and  in  many  schools  there  is  not  even  a 
nominal  text-book,  so  that  the  personality  of  the  instructor  counts 
for  everything.  However,  many  exercise  books  are  used. 

The  question  of  a  course  preliminary  to  geometric  instruction 
is  not  defined  in  the  official  curriculum.  The  consensus  of 
opinion  among  the  instructors  is  as  follows.  It  should  begin  in 
the  Quinta  and  continue  through  the  Quarta  as  a  purely  pro- 
paedeutic course.  The  course  should  start  from  the  contempla- 
tion of  the  solids,  seek  out  the  fundamental  properties,  and  de- 
velop these  and  give  them  definite  shape.  There  should  be  funda- 
mental exercises  in  the  use  of  the  compasses,  straightedge,  and 
protractor.  Although  a  minority  advocates  a  formal  geometry 
from  the  start,  with  definitions  and  demonstrations,  the  majority 
opposes  formal  proofs  before  the  Untertertia. 

Models  are  in  general  use,  especially  in  the  upper  classes,  where 
they  find  frequent  application  in  stereometry  and  descriptive 
geometry.  They  are  usually  made  by  the  class,  or  by  the  more 
capable  pupils.  The  author,  Dr.  Wirz,  wants  the  "  inner  sight  " 
developed  and  hence  he  does  not  think  models  at  all  necessary. 
In  fact,  he  feels  that  models  do  more  harm  than  good,  because 
pupils  frequently  become  satisfied  with  empirical  proofs. 

In  regard  to  original  exercises,  most  teachers  oppose  giving 
them  before  the  Untertertia,  for  they  take  too  much  of  the  pupils' 
time  without  help,  and  the  instructors  have  very  little  time  to 
make  helpful  corrections  for  a  large  class.  Some  think  that 
practical  exercises  should  be  the  rule  in  mathematics,  while  others 
wish  the  aim  to  be  entirely  logical.  The  author  points  out  that 
historically  the  greatest  advances  were  made  when  mathematics 
was  treated  independently. 

The  individual  opinions  regarding  the  function-concept  vary 
greatly,  but  the  majority  hold  a  compromise  position.  The 
younger  teachers  advocate  it  most  strongly,  but  it  remains  a 
fact  that  the  function-concept  has  not  made  much  headway  in 
Elsass-Lothringen.  As  regards  the  calculus,  most  teachers  favor 
it  for  the  Oberrealschule  only,  but  desire  a  knowledge  of  the 
symbolism  and  fundamental  methods  for  the  Gymnasium  also. 

There  is  a  great  demand  for  lessening  the  routine  work  in 
the  formal  operations,  so  as  to  devote  more  time  to  geometry. 


60  Teachers  College  Record  [126 

This  is  opposed,  and  justly,  by  those  who  assert  that  the  analytic 
power  in  mathematics  is  just  as  important  as  the  synthetic  power, 
if  not  more  so.  The  cry  for  combining  planimetry  and  stereom- 
etry is  opposed  by  a  great  majority  on  both, psychological  and 
logical  grounds.  Descriptive  geometry  is  being  generally  re- 
quired. 

In  the  last  decade  great  interest  has  been  shown  in  the  history 
of  mathematics,  and  its  introduction  in  the  last  year  of  the  Gym- 
nasium is  being  advocated  by  all  mathematicians.  The  question 
as  to  what  mathematics  should  be  taught  in  the  upper  classes 
is  a  matter  of  considerable  discussion.  There  is  the  danger  of 
forcing  too  much  work  upon  those  who  have  no  liking  or  apti- 
tude for  it,  and,  on  the  other  hand,  of  giving  a  course  too 
mediocre  for  the  talented  student.  The  best  suggestion  seems 
to  be  the  division  of  upper  classes  into  two  groups,  the  first  that 
of  mathematics  and  science,  and  the  second  that  of  history  and 
language. 

For  the  Reifeprufung,  described  above,  the  following  changes 
are  advocated :  (a)  The  same  problems  in  all  the  schools ;  (b) 
the  choice  of  one  large  problem  instead  of  four;  (c)  no  oral 
examination  in  some  of  the  subjects ;  (d)  written  examinations 
to  be  made  easier  in  certain  subjects. 

Most  of  the  problems  which  confront  Elsass-Lothringen  are 
the  same  as  our  problems,  and  it  behooves  us  to  consider  what 
changes  should  be  made  with  the  same  deliberation  that  is  shown 
in  this  part  of  Germany.  One  thing  that  seems  to  be  invariable 
in  all  the  German  states  is  the  fact  that  some  form  of  geometry 
is  taught  before  any  very  serious  work  is  undertaken  in  algebra. 
This,  however,  is  not  to  be  misunderstood,  for  "Arithmetik  " 
includes  our  elementary  algebra. 

THE  REQUIREMENTS  FOR  AND  TRAINING  OF  TEACHERS  IN 
MATHEMATICS 

The  requirements  for  the  license  to  teach  mathematics  are : 
I.  Pure  mathematics. 

(a)  For  first  degree  (license  to  teach  classes  up  to  the  Unter- 
secunda)  :  sound  knowledge  of  elementary  mathematics,  knowl- 


I2/]     The  Present  Teaching  of  Mathematics  in  Germany      61 

edge  of  plane  analytic  geometry,  conic  sections,  and  the  funda- 
mental principles  of  the  integral  and  differential  calculus. 

(b)  For  second  degree  (all  classes)  :  in  addition  to  above,  a 
familiarity  with  higher  geometry,  arithmetic,  algebra,  higher 
analysis,  and  analytic  mechanics,  including  the  ability  to  solve 
problems  that  are  not  too  difficult. 

II.  Applied  mathematics.  A  knowledge  of  descriptive  geom- 
etry up  to  and  including  central  projection,  familiarity  with 
mathematical  methods  connected  with  technical  mechanics,  with 
graphical  statics,  and  with  geodesy  or  astronomy. 

In  addition  to  this  examination  in  mathematics,  the  license  for 
first  degree  requires  the  taking  of  an  examination  in  one 
other  subject,  and  the  license  for  second  degree  in  two. 
The  general  examination  taken  by  all  (philosophy,  pedagogy, 
German,  Latin,  and  religion)  is  held  before  the  Wissenschaftliche 
Kommission  in  Strassburg,  where  most  of  the  candidates  go  for 
their  higher  education. 

The  practical  pedagogical  training  of  the  candidates  has  been 
the  same  since  1871,  and  was  taken  from  the  Prussian  system. 
This  consists  in  passing  the  Probejahr  (trial  year). 

1 i )  6-8  hours  teaching  every  week. 

(2)  Attendance    for    observation    in    classes    of    the    major 
subject. 

(3)  Conferences  and  seminar  (the  Probe-seminar). 

(4)  Theme  at  end  of  the  year,  the  subject  being  assigned  in 
the  first  semester.     The  candidate  must  show  ability  to  attack  a 
practical  problem  and  to  look  up  the  literature  relating  to  it. 
If  the  candidate  passes  his  Probejahr  he  receives  a  certificate. 
He  then  enters  the  system  immediately,  although  some  prefer 
to  take  an  extra  year  as  a  practice  teacher. 

The  objections  to  this  system  are  that  accidents  play  too  great 
a  part,  and  that  there  is  entirely  too  much  emphasis  on 
the  study  of  methods  of  teaching.  Although  some  study  of 
methods  is  necessary  to  give  training  in  the  grouping  of 
material,  and  in  getting  different  points  of  view,  still  to  the 
question  as  to  whether  pedagogic  training  is  of  much  aid.  the 
answer  "  No  "  is  more  often  given  than  any  other.  No  inves- 


62  Teachers  College  Record  [128 

tigation  of  the  work  of  the  other  states  is  required  as  in  Prussia 
and  other  parts  of  Germany.  Only  once  (1901)  was  an  investi- 
gation into  the  schools  of  the  other  German  states  made.  A 
system,  definite  for  all,  like  the  Prussian,  would  be  a  good  thing 
for  Elsass-Lothringen. 

If  we  compare  the  requirements  for  the  high-school  teacher 
of  mathematics  in  our  country  with  those  of  Elsass-Lothringen, 
we  see  that  in  pure  irathematics  the  requirements  for  the  first 
degree  in  that  province  are  about  the  same  as  those  for  college 
graduation  here  and  for  license  to  teach  in  our  better  high 
schools,  except  in  places  like  New  York  where  they  are  slightly 
higher.  We  have  absolutely  no  requirement  in  applied  mathe- 
matics, while  the  general  examination  in  pedagogy  is  not  at  all 
comparable  to  the  general  examination  in  Elsass-Lothringen,  or 
for  that  matter  in  any  German  province.  The  requirements  for 
the  second  degree  in  Elsass-Lothringen  would  leave  very  few 
eligible  teachers  in  this  country.  We  may  conclude,  therefore, 
that  one  of  the  reasons  why  Germany  can  accomplish  better 
results  in  mathematics  than  America,  is  the  fuller  and  more 
thorough  equipment  of  her  teachers. 


CHAPTER  IX 
MATHEMATICS  IN  GERMAN  TECHNICAL  SCHOOLS1 

Donald  T.  Page 

This  article  is  an  attempt  to  state  briefly  the  main  points  brought 
out  in  a  report  recently  published  by  Dr.  Jahnke  of  Berlin  on 
the  mathematics  of  the  higher  industrial  schools.1  The  report 
deals  with  the  special  schools  of  mining,  military  science,  fores- 
try, agriculture,  and  commerce.  It  gives  a  brief  historical  state- 
ment concerning  each  of  the  large  schools,  and  a  description  of 
the  courses  in  mathematics,  with  their  changes  and  development. 
It  also  treats  of  the  need  for  mathematics  in  the  requirements 
for  the  post  and  telegraph  service,  and  mentions  briefly  several 
institutions  that  offer  courses  in  higher  mathematics. 

The  various  schools  exhibit  individual  characteristics;  but  in 
the  ninety  years  since  technical  institutions  began  to  develop 
there  has  grown  up  a  strong  movement  against  pure  mathematics, 
which  has  affected  the  courses  in  most  of  the  schools.  This 
movement  has  been  caused  by  a  demand  for  practical  problems, 
and  its  effect  is  shown  in  the  new  text-books,  and  in  the  remod- 
eling of  courses  so  that  the  calculus  and  mechanics  are  taught 
in  one  course  under  the  name  "  Higher  Mathematics  and 
Mechanics." 

Opposition  to  this  movement  is  felt  by  many  of  the  best  engi- 
neers who  assert  that  an  engineer  must  have  a  thorough  knowl- 
edge of  mathematics  and  that  the  calculus  and  the  theory  of  equa- 
tions are  especially  valuable  in  developing  thinking  ability. 

THE  SCHOOLS  OF  MINING 

Most  of  the  schools  of  mining  have  the  same  entrance  require- 
ments as  the  universities,  and  some  of  them  have  an  extra  re- 


*Die   Mathematik    an    Hochschulen    fur  besondere    Fachgebiete,   von 

Dr.    E.    Jahnke,    Etatsmassigem    Professor  an    der    Kgl.    Bergakademie, 
Berlin,  Leipzig  and  Berlin,  1911. 
129]  63 


64  Teachers  College  Record  [130 

quirement  of  at  least  one  year  of  practical  work.  The  course 
is  usually  three  years  in  length.  In  Prussia  many  of  the  gradu- 
ates take  civil  service  examinations.  The  mathematics  taught  in 
the  Berlin  Academy  illustrates  fairly  well  the  ground  covered, 
and  is  given  in  the  following  table : 


1st  year 

2nd  year 

3rd  year 

Analytic  Geometry, 
Algebra,  Descriptive 
Geometry 

Higher  Mathematics 
and  Mechanics 

Theory  of  Errors  and 
Method  of  Least 
Squares 

The  use  of  models  and  instruments  is  encouraged,  while  part 
of  the  allotted  time  is  reserved  for  the  solution  of  practical 
problems.  Freiburg  Academy  (1765)  was  one  of  the  first  schools 
to  advocate  vocational  education.  The  course  is  four  years  in 
length  and  has  not  been  as  much  affected  by  the  movement 
against  mathematics  as  the  courses  in  other  schools. 

MILITARY  SCHOOLS 

In  the  military  schools  there  is  a  growing  emphasis  upon  ap- 
plied mathematics.  A  description  of  the  course  at  the  Berlin 
Military  Academy  will  give  a  general  idea  of  the  mathematics 
offered  in  these  schools.  At  Berlin  the  course  is  three  years 
in  length.  The  entrance  requirements  are  plane  geometry,  algebra 
through  quadratic  equations,  logarithms,  and  plane  trigonometry. 
The  first  year's  course  includes  spherical  trigonometry,  plane 
analytic  geometry,  part  of  the  differential  calculus,  and  infinite 
series.  The  second  year's  work  completes  the  differential  and 
integral  calculus,  and  takes  solid  analytic  geometry,  some  prob- 
lems in  analytic  mechanics,  and  some  astronomy.  In  the  third 
year  surveying  and  more  advanced  astronomy  are  taken. 

FORESTRY  SCHOOLS 

In  some  of  the  forestry  schools  the  course  is  three  years  long, 
in  others  four.  In  some,  an  apprenticeship  of  several  months 
to  a  chief  forester  is  required,  in  addition  to  the  general  re- 
quirement of  certificate  for  entrance  to  a  university.  The  for- 
estry schools  in  South  Germany  offer  more  courses  in  pure 


13 1  ]     The  Present  Teaching  of  Mathematics  in  Germany      65 

mathematics  than  do  those  in  North  Germany.  The  mathematics 
courses  generally  include  the  calculus,  geodesy,  plan  drawing, 
forest  computations,  and  statistics. 

SCHOOLS  OF  AGRICULTURE 

The  schools  of  agriculture  give  few  courses  in  mathematics, — > 
usually  surveying,  levelling,  and  plan-drawing. 

SCHOOLS  OF  COMMERCE 

The  mathematics  in  the  schools  of  commerce  usually  consists 
of  commercial  and  political  arithmetic ;  but  courses  in  higher 
mathematics  are  occasionally  offered. 

POST  AND  TELEGRAPH  SERVICE 

The  requirements  for  entrance  to  the  Post  and  Telegraph  Ser- 
vice do  not  include  mathematics ;  but  schools  that  prepare  candi- 
dates have  frequently  offered  courses  in  analytic  geometry,  the 
calculus,  differential  equations,  and  mathematical  physics.  Higher 
mathematics  has  proved  helpful  to  telegraph  engineers  who  have 
studied  it,  and  is  therefore  urged  as  a  needed  addition  to  the 
present  requirements. 

Industrial  conditions  in  Germany  are  such  that  the  need  for 
both  pure  and  applied  mathematics  is  increasing.  Mathematicians 
are  far  behind  in  the  solution  of  problems  raised  in  physics  and 
engineering.  These  conditions  seem  likely  to  bring  about  a 
strengthening  of  the  present  position  of  mathematics  and  a  weak- 
ening of  the  opposition  to  it. 


CHAPTER  X 
THE  GERMAN   MIDDLE  TECHNICAL  SCHOOLS1 

Miriam  E.  West 

This  report  concerning  mathematics  instruction  in  the  German 
middle  technical  schools  of  the  machine  industry  consists  of 
six  chapters  which  treat  of  the  development  of  these  schools, 
their  organization,  the  mathematics  instruction  given  in  them, 
the  text-books,  the  method  of  treatment  of  the  different  subjects 
of  instruction,  and  the  preparation  of  the  teacher  of  mathematics. 

Public  technical  education  in  the  schools  of  Germany  is  a 
product  of  the  nineteenth  century.  After  the  invention  of  the 
steam  engine,  when  technical  instruction  was  flourishing  in 
France,  there  came  a  demand  in  Germany  for  technical  train- 
ing. Peter  Wilhelm  Beuth  was  the  founder  of  public  technical 
instruction  in  Prussia.  Between  1817  and  1821  four  schools 
were  established  in  Germany,  and  within  a  very  short  time 
eighteen  more.  The  majority  of  these  schools  had  a  one-year 
course,  but  the  institution  in  Berlin,  and  several  others,  had  two- 
year  courses.  The  mathematical  instruction  of  the  Prussian 
schools  included  geometry,  arithmetic,  algebra  through  quadratic 
equations,  a  little  stereometry,  and  a  little  trigonometry.  In  1850, 
there  were  added  the  principal  properties  of  conies,  and  theo- 
retical and  practical  surveying.  In  1870  a  reform  was  instituted. 
The  schools  were  made  three-year  schools,  and  in  the  two  lowest 
classes  general  instruction  in  the  languages,  history,  geography, 
and  the  sciences  was  given.  In  the  third  class  the  instruction 
was  divided  into  four  sections : 

(a)  Preparation  for  entrance  to  higher  technical  schools. 

(b)  Preparation  for  the  building  trade. 


1  Der  Mathematische  Unterricht  an  den  Deutschen  Mittlern  Fachschulen 
der  Maschinenindustrie,  von  Dr.  Heinrich  Griinbaum,  Lehrer  der  Mathe- 
mamatik  und  Physik  am  Rheinischen  Technikum  in  Bingen.  July,  1910. 

66  [132 


133]     The  Present  Teaching  of  Mathematics  in  Germany      67 

(c)  Preparation  for  technical  mechanical  work. 

(d)  Preparation  for  technical  chemistry. 

The  mathematics  course  was  broadened  to  include  the  follow- 
ing subjects:  determinants,  combinations,  binomial  theorem,  com- 
putation of  logarithms  and  trigonometric  functions  by  infinite 
series,  continued  fractions  with  applications,  spherical  trigonome- 
try, the  elements  of  analytic  geometry.  The  completion  of  a  five- 
years'  course  in  a  Realschule  was  required  for  entrance  to  these 
schools.  A  number  of  Realschulen,  at  this  time,  offered  this 
technical  education  by  adding  to  their  five  years  two  more  of 
general  instruction  and  one  year  of  vocational  instruction.  These 
became  later  the  Oberrealschulen.  This  general  education  with 
only  one  year  of  technical  instruction  proved  unsatisfactory,  so 
in  1889  the  Society  of  German  Engineers  took  the  matter  in 
hand.  As  a  result  the  middle  technical  schools  were  established, 
offering  two  years  of  vocational  training.  Later  another  half 
year  was  added,  making  five  half  year  classes.  They  required 
for  entrance,  besides  the  completion  of  the  work  of  a  Realschule, 
at  least  two  years  practice  at  some  vocation. 

There  are  about  forty  institutions  in  Germany  which  may  be 
classed  as  middle  technical  schools.  Some  of  these  are  state 
schools;  others  are  private  or  town  schools.  Of  these  the 
Prussian  Maschinenbauschulen  are  controlled  by  the  state,  and 
the  final  examinations  are  given  by  the  royal  examination  com- 
mission. For  entrance  to  these  schools  is  required  the  "  Einjahr- 
igen-Zeugnis "  (a  certificate  which  exempts  the  holder  from 
one  year  of  military  service),  seven  years  in  a  higher  school, 
the  completion  of  the  work  of  a  Volkschule  and  work  at  a 
preparatory  school,  or  the  passing  of  an  examination  in  the  fol- 
lowing subjects:  German,  arithmetic,  algebra  through  quadratic 
equations  and  logarithms,  plane  geometry,  trigonometry  through 
the  computation  of  right-angled  triangles,  and  stereometry.  The 
satisfactory  completion  of  the  course  in  these  schools  qualifies 
one  to  enter  the  following  branches  of  public  service : 

(a)  State   railroad  service,  as   officers,   superintendents,   en- 
gineers, etc. 

(b)  Marine  service. 

(c)  Royal-Bureau  for  the  construction  of  artillery. 


68  Teachers  College  Record  [134 

Similar  to  these  Prussian  schools  are  the  technical  schools  of 
Hamburg,  Bremen  and  Niirnberg  and  the  royal  Fachschule  of 
Wurzburg.  These  are  state  schools  and  are  included  in  the 
forty  mentioned  above.  They  have  a  more  elaborate  organiza- 
tion than  the  Prussian  Maschinenbauschulen.  Hamburg  pre- 
pares for  five  distinct  vocations:  machine  construction,  elec- 
trical engineering,  ship  building,  construction  of  ship  machinery, 
and  ship  engineering.  Some  of  the  other  schools  have  a  larger 
variety  of  courses,  and  some  fewer.  The  private  schools,  since 
they  are  not  controlled  by  the  state,  can  adapt  themselves  more 
readily  to  the  needs  of  the  various  vocations,  and  many  of  them 
keep  up  with  the  changes  in  industry  better  than  the  state 
schools. 

The  pupils  who  enter  the  middle  technical  schools  may  be 
divided  into  two  groups.  Those  of  the  first  group  possess  the 
"  Einjahrigen  Zeugnis,"  have  usually  completed  the  six  classes 
of  a  higher  school,  and  have  had  one  or  two  years  of  practice 
in  a  workshop.  The  pupils  of  the  other  group  possess  only  a 
Volkschule  education  and  have  had  longer  practice.  Mean- 
while they  have  increased  their  knowledge  in  industrial  mathe- 
matics by  attendance  at  some  continuation  school.  In  some 
schools  the  pupils  of  the  second  group  are  put  in  a  preparatory 
department,  while  in  others  the  pupils  entering  are  divided  into 
two  sections  according  to  their  preparation. 

The  following  is  the  mathematical  curriculum  in  the  Prussian 
Maschinenbauschulen : 

I.  ALGEBRA:   powers,  roots,  logarithms,  equations  of  the  first 
degree,  exponential  equations,  arithmetic  and  geometric  series, 
interest  and  annuities,  convergence  and  divergence  of  infinite 
series,   binomial    theorem,   exponential    and   logarithmic    series, 
natural  logarithms,  maxima  and  minima,  graphical  solution  of 
numerical  equations. 

II.  PLANE  GEOMETRY:    the  most  important  propositions  of 
elementary  geometry,  circle,  area  of    plane  figures,  proportion, 
construction  problems. 

III.  TRIGONOMETRY:  trigonometric  functions   and  their  rela- 
tions to  each  other,  use  of  trigonometric  tables,  computation  of 
triangles,  quadrilatrals,  and  regular  polygons. 


135]     The  Present  Teaching  of  Mathematics  in  Germany      69 

• 

IV.  STEREOMETRY:  straight  lines  and  planes  in  space,  the  tri- 
hedral angle,  the  regular  solids;  surfaces  and  volumes  of  the 
prism,  pyramid,  cylinder,  cone,  sphere,  truncated  solids,  parts 
of  the  sphere;  general  methods  for  computation  of  solids,  such 
as  Guldin's  and  Simpson's  rules;  application  to  computation  of 
volume  and  weight. 

Many  of  the  schools  have  introduced  courses  in  differential 
and  integral  calculus.  This  is  especially  true  of  the  private 
schools. 

The  aim  of  instruction  in  the  middle  technical  schools  is  dif- 
ferent from  that  in  the  higher  or  so-called  cultural  schools. 
In  the  former  mathematics  is  studied  from  a  practical  stand- 
point, and  as  a  tool  for  solving  the  technical  problems  of  the 
various  vocations.  It  is,  therefore,  impossible  to  borrow  the 
methods  of  the  higher  schools,  in  which  mathematics  is  usually 
studied  as  an  end  in  itself.  A  hundred  years  of  vocational 
schools  has  not  been  sufficient  to  create  methods  peculiarly 
adapted  to  technical  instruction.  The  text-books  used  in  the 
two  schools  are  largely  the  same.  The  fact  that  the  instruction 
is  similar  in  the  two  schools  is  due  partly  to  the  fact  that  many 
of  the  vocational  schools  were  originally  organized  in  connec- 
tion with  higher  schools.  The  pupils  in  these  schools  are  older 
than  those  in  the  higher  schools,  being  from  eighteen  to  twenty- 
two  years  of  age. 

The  instruction  in  the  technical  schools  has  or  should  have 
certain  distinguishing  features.  The  problems  in  algebra,  so  Dr. 
Griinbaum  asserts,  should  be  chiefly  practical  ones,  for  which 
geometry,  mechanics,  and  physics  offer  a  good  field.  In  order 
to  obtain  the  required  skill  it  is  necessary  for  the  pupil  to  work 
many  problems,  but  it  is  of  no  great  value  for  him  to  work  those 
for  which  he  will  later  have  tables.  Practice  in  use  of  the  slide 
rule  and  shortened  methods  of  computation  as  well  as  practice 
in  drawing  is  considered  important. 

In  the  teaching  of  mathematics  there  is  danger  of  too  much 
emphasis  being  placed  upon  the  use  of  formulas.  If  mathe- 
matical principles  are  dried  up  to  formulas  and  the  only  mental 
work  consists  in  applying  them  to  certain  particular  examples, 
the  natural  result  is  that  the  principles  behind  the  formulas  are 


7O  Teachers  College  Record  [136 

soon  forgotten,  and  the  pupil  has  simply  a  collection  of  rules, 
with  no  knowledge  of  their  application.  It  is  important  to  keep 
the  principles  in  mind.  The  pupil  need  not  be  expected  to  re- 
member a  large  body  of  formulas.  The  more  simple  ones  he 
can  derive  easily  for  himself,  and  for  the  more  difficult  ones 
which  have  been  previously  derived  in  class  he  may  have  access 
to  a  book  containing  a  collection  of  them. 

In  algebra  the  most  important  thing  for  the  pupil  to  acquire 
in  the  first  few  weeks  is  skill  in  solving  equations.  The  other 
elementary  instruction  in  algebra  should  be  made  of  service  to 
this  fundamental  problem.  The  fundamental  operations,  factor- 
ing, and  fractions  should  be  taught  with  this  in  view.  On  this 
skill  will  depend  ease  in  development  of  formulas.  One  notices 
the  dependence  of  these  schools  on  the  higher  schools  in  the 
use  of  the  old  traditional  problems,  —  the  boy  with  the  nuts,  the 
farmer's  wife  with  the  eggs,  the  tank  with  the  pipes,  etc.  The 
real  technical  problems  are  seldom  found  in  the  text-books.  The 
treatment  of  powers,  roots,  and  logarithms  is  pronounced  by  all 
to  be  too  formal.  There  is  too  much  dependence  on  the  rules 
concerning  exponents  and  indices. 

In  many  of  the  text-books  great  stress  is  placed  upon  the 
solution  of  the  radical  equation.  Yet  the  text-book  writer  often 
fails  to  notice  that  values  of  x  obtained  do  not  satisfy  the  equa- 
tions. For  example: 


V  36  +  x 
36  +  x  =  324  +  36  V  x  -f-  x 

V~x"  =  —  8 


A  graphic  representation  of  the  parabolas  y2  =  36  +  x  and 
(y—  i8)2  =  x  would  make  clear  the  difficulty  in  this  solution. 

The  function  concept  has  penetrated  very  little  into  the  in- 
struction. Such  concepts  as  those  of  constant,  dependent  and 
independent  variable,  and  function  are  of  great  value  in  the 
solution  of  equations  and  in  the  use  of  formulas.  There  is  still 
a  large  space  in  the  text-book  devoted  to  the  old  style  propor- 
tion. If  this  were  put  in  the  field  of  the  function  concept,  it 


137]     The  Present  Teaching  of  Mathematics  in  Germany      71 

would  be  fruitful.     For  proportion  only  y  =  ax  is  needed,  and 
not  the  form  yx :  y2  =  xt :  x2. 

In  the  solution  of  equations  of  higher  degree  an  average 
course  would  give  the  remainder  theorem  and  the  theorem  con- 
cerning the  relationship  between  the  coefficients  and  roots  of  an 
equation  of  the  nth  degree.  This  gives  a  method  for  finding 
the  integral  roots.  The  most  common  solution  of  the  equation 
f  (x)  =  o  is  by  finding  the  intersection  of  its  curve  with  the 
x  =  axis  or  by  the  intersection  of  the  two  curves 

y  =  m  (x), 
and     y  =  n  (x), 
if  m(x)— n(x)— f  (x). 

Graphical  methods  are  used  for  the  solution  of  cubic  and  bi- 
quadratic equations.  There  is  little  practical  need  for  the  solu- 
tion of  equations  of  higher  degrees,  difficult  quadratic  equations 
with  several  unknown  quantities,  or  reciprocal  equations. 

In  geometry,  the  Euclidean  method  of  proof  is  used  very  little. 
Symmetry  is  an  important  means  of  proof.  The  propositions 
concerning  parallel  lines  are  proved  through  motion.  In  all  the 
new  books  many  of  the  construction  problems  are  giving  way 
to  problems  in  computation  of  figures.  The  introduction  of  the 
ideas  of  kinematics  into  plane  geometry  is  to  be  commended. 
Through  these  should  come  the  explanation  of  the  ideas  of  trans- 
lation, rotation,  rolling,  and  the  simple  propositions  concerning 
the  moment  of  force.  The  study  of  solid  geometry  in  connection 
with  plane  geometry  is  of  advantage  tq  the  pupil  in  the  tech- 
nical school,  for  he  has  as  much  need  of  the  former  as  of  the 
latter,  and,  by  connecting  them,  he  is  able  to  transfer  the  ideas 
of  plane  geometry  to  the  figures  in  space.  The  instruction  in 
stereometry  should  not  be  simply  the  application  of  formulas 
to  problems  in  computation  of  surfaces,  areas,  and  volumes,  but 
should  serve  above  all  to  strengthen  the  idea  of  space.  More- 
over it  should  prepare  for  possible  applications.  This  can  be 
attained  if  the  pupil  not  only  computes  the  area  and  volume 
of  the  regular  solids  but  is  accustomed  to  compute  other  solids 
by  breaking  them  up  by  means  of  planes  into  elementary  bodies. 

In  plane  trigonometry  the  instruction  is  similar  to  that  in  the 
higher  schools,  but  many  are  of  the  opinion  that  such  thorough 


72  Teachers  College  Record  [138 

treatment  is  unnecessary.  All  practical  need"  would  be  satisfied 
if  a  detailed  handling  of  the  right  angle  triangle  and  its  applica- 
tions, together  with  the  sine  and  cosine  propositions,  were  given. 
A  review  of  the  text-books  shows  that  mechanics  and  physics 
are  rich  in  examples  which  require  the  solution  of  the  right 
angle  triangle.  But  for  the  scalene  triangle  it  is  with  difficulty 
that  the  authors  find  practical  problems.  Great  exactness  is  not 
necessary  in  practical  work,  so  logarithmic  solutions  are  not  im- 
portant. 

In  analytic  geometry  the  function  which  is  of  the  greatest  im- 
portance is  y  =  ax2  -f-  bx .+  c.  Of  the  higher  curves,  y  =  ax°,  the 
entire  function  of  the  nth  degree,  the  exponential  curve,  the 


2    TT 


curves  y  =  sin  x  and  y  =  A  sin   — (x  —  6),    the   cycloid    and 

several  spirals  (in  polar  coordinates),  are  the  principal  ones 
considered.  The  solid  analytic  geometry  is  based  on  the  plane 
analytic  geometry.  The  direction  cosines  of  a  line  and  the 
geometric  significance  of  f  (x,  y,  z)=o  should  be  familiar 
to  the  pupils.  The  pupils  comprehend  easily  and  with  interest 
those  equations  of  surfaces  which  are  closely  connected  with 
the  equations  of  well-known  curves,  as: 


a       b 

x2  ,  y2  ,  /z'\ 

I   =  i 

a2      b1     \c2/ 

x2+y2  +  (z2)  =r2,  etc. 

The  fact  that  differential  and  integral  calculus  has  not  been 
introduced  into  the  schools  of  Prussia  is  due  mainly  to  one  man, 
Professor  Gustav  Holzmuller,  who  has  persistently  fought  its 
introduction.  He  would  have  the  problems  which  would  natur- 
ally be  solved  by  calculus  solved  by  the  so-called  elementary 
method.  In  his  text-book,  "  Ingenieur  Mathematik,"  he  has 
the  following: 

Let  the  equation  of  the  curve  be 

i          2      "  3  4 


139]     The  Present  Teaching  of  Mathematics  in  Germany      73 

To  find  the  tangent  to  this  curve  which  makes  a  certain 
angle  a  with  the  y  axis,  the  following  formula  is  used: 

tan  a  =  a  +  by  +  cy2  +  dy8. 
Likewise  given  the  equation 

q  =  a  +  by  -f-  cy2  +  dy3, 

q  being  the  slope  of  the  tangent  to  the  curve  at  any  point,  the 
equation  of  the  curve  is 

x=^y+by!  +  sz!  +dy' 

1  .3         4 

How  does  this  differ  from  differentiation  and  integration?     In 

the  same  elementary  manner  he  develops  the  general  binomial 
theorem,  exponential  and  logarithmic  series,  the  sine  and  cosine 
series.  For  the  summation  of  the  integral  [  x"  dx  it  is  neces- 
sary, if  this  method  is  used,  to  know  the  sum  of  the  powers 

in  +  2n  +  3n     .     .     .     mn  =  2  x". 

i 

These  can  be  found  by  this  identity 

(a+i)n  =  an+nant1+ 

putting  a  =  o,  i,  2, m  and  adding  all  the  equations. 

If  this  is  continued  for  n  =  2,  3,  4, ,  the  formula  is 

obtained  2  x2,  2  x3,  2  x4,  etc.  Contrast  this  with  the  general 
interpretation  of  2  x"  by  integration.  In  a  similar  manner  for- 
mulas are  obtained  for  the  areas  and  volumes  of  solids.  These 
problems,  which  until  recently  have  been  treated  by  this  elemen- 
tary method,  are  now  coming  to  be  handled  by  the  method  of 
the  calculus,  which  is  much  simpler  and  has  fewer  formulas. 

The  time  given  to  mathematics  need  not  be  increased  for  the 
introduction  of  the  calculus,  since  the  calculus  merely  affords  a 
new  method  for  solving  old  problems.  After  the  idea  of  differ- 
ential quotients  is  made  clear  and  the  simple  functions  and  their 
combinations  are  differentiated,  then  follows  the  discussion  of 
curves,  maxima  and  minima,  turning  points,  slope,  rate,  accelera- 
tion, and  infinite  series.  After  an  introduction  to  integration  the 
entire  remaining  time  can  be  given  to  the  important  technical 
computations,  (surfaces,  solids,  moments  of  inertia,  centers  of 
gravity.)  Books  are  now  being  published  which  strive  to  supply 
the  need  of  the  technical  schools  in  this  direction. 


74  Teachers  College  Record  [140 

Up  to  the  present  time  the  required  preparation  for  a  teacher 
in  a  middle  technical  school  has  been  the  graduation  from  a  nine- 
class  school,  together  with  four  years  academic  training.  It  is 
necessary  that  the  teacher  have  a  broad  view  of  the  need  of 
mathematics  in  the  technical  callings.  The  question  which  as 
yet  has  not  been  settled  is,  where  can  the  teacher  obtain  this 
broad  view?  If  he  goes  to  the  University  he  will  not  get  it, 
although  several  courses  are  now  being  offered  at  Gottingen 
which  help  to  meet  this  need,  such  as  general  applied  mathe- 
matics, mathematical  instruction  in  higher  schools,  technical 
mechanics.  It  would  seem  as  though  the  technical  high 
schools  would  offer  the  necessary  preparation  for  the  teacher. 
But  if  he  is  trained  here,  he  is  obliged  to  specialize  along  one 
line  and  thus  he  fails  to  get  a  view  of  the  needs  of  other  lines. 
The  technical  high  schools  cannot  afford  to  offer  special  courses 
for  teachers,  since  the  demand  in  this  line  is  so  small,  only 
about  six  being  needed  each  year.  The  training  is  still  an  un- 
settled problem.  The  engineer  lacks  the  necessary  training  for 
a  teacher,  and  the  teacher  trained  in  pure  mathematics  lacks  the 
necessary  knowledge  of  mathematics  as  applied  to  mechanics. 


CHAPTER  XI 

MATHEMATICS  IN  GERMAN  SCHOOLS  OF 
NAVIGATION1 

Donald  T.  Page 

In  early  times  there  was  little  need  for  schools  of  navigation. 
The  young  sailors  could  learn  the  most  important  requirements 
by  experience  on  board  ship,  and  could  learn  from  the  older 
sailors  what  few  scientific  principles  were  necessary.  The  leisure 
time  on  a  voyage  and  during  the  winter  season  was  employed 
to  advantage  in  such  studies. 

As  commerce  grew,  ships  of  larger  size  were  built,  voyages 
became  longer,  and  better  trained  seamen  were  needed.  During 
the  eighteenth  century  this  need  of  trained  seamen  was  felt  in 
the  trade  with  the  East  Indies  and  in  the  great  growth  of  trade 
at  the  close  of  the  War  for  Independence  in  America.  To  satisfy 
the  demand,  private  schools  sprang  up. 

Each  private  school  had  its  own  standards  of  excellence.  In 
many  cases  the  student  gave  his  teacher  a  certificate  of  com- 
mendation in  exchange  for  his  certificate  of  graduation.  The 
school  at  Hamburg  rendered  valuable  service  for  many  years  by 
publishing  a  nautical  almanac. 

In  1793  Emden  made  the  requirement  that  candidates  for 
positions  of  mate  or  captain  must  pass  an  examination.  The 
other  states  soon  made  similar  requirements;  but  certificates  of 
one  state  were  not  accepted  in  another  state,  and  there  was  no 
uniformity  in  the  requirements  until  1870,  when  the  Bundesrat 
instituted  the  present  system. 

After  the  examinations  were  made  compulsory,  public  schools 
of  navigation  were  finally  organized.  There  were  special  schools 
established  for  the  education  of  captains  of  small  craft,  and  six 


*Der  Mathematische  Unterricht  an  den  Deutschen  Navigationsschulen, 
von  Dr.  C.  Schilling  und  Dr.  H.  Melddau,  Leipzig  and  Berlin,  1911. 

141]  75 


76  Teachers  College  Record  [142 

such  schools  are  now  in  existence.  These  also  prepare  for 
entrance  to  the  higher  schools  with  which  this  article  is  chiefly 
concerned. 

There  are  eighteen  of  these  higher  schools  offering  courses 
leading  to  examinations  which  qualify  for  the  position  of  mate 
or  captain  of  large  ships.  There  are  no  entrance  requirements. 
The  purpose  of  the  course  is  to  prepare  for  examination,  and 
for  this  reason  the  subject  matter  and  methods  of  study  are 
somewhat  limited.  The  course  for  mates  is  about  eight  months, 
and  for  captains  about  four  months ;  but  at  least  two  years'  ser- 
vice must  intervene. 

Examinations  are  in  charge  of  an  examination  commission  at 
each  school,  but  are  provided  for  in  such  a  way  as  to  give 
uniformity  throughout  the  country.  To  be  eligible  for  the  mate's 
examination,  a  candidate  must  have  had  at  least  forty-five  months' 
experience  after  reaching  fifteen  years  of  age.  This  experience 
must  include  twenty-four  months'  service  as  able  seaman  on  a 
merchant  vessel  and  twelve  months  on  a  sailing  vessel.  For  the 
captain's  examination,  a  candidate  must  have  served  twenty-four 
months  as  mate,  and  must  present  notes  and  computations  show- 
ing that  he  has  had  practice  in  taking  nautical  observations. 
These  requirements  bring  the  age  of  candidates  up  to  22  years 
for  mates  and  26  years  for  captains.  During  a  recent  year 
there  were  examined  746  of  the  former  and  441  of  the  latter. 

The  examination  consists  of  three  parts,  oral,  practice,  and 
written.  In  order  to  preserve  uniformity  the  most  weight  is 
given  to  the  written  part,  which  consists  of  sets  of  problems 
divided  into  three  groups  of  seven  subjects.  Out  of  a  supply 
of  problems,  packets  are  made  up  so  that  each  candidate  re- 
ceives one  problem  on  each  of  the  twenty-one  subjects.  The 
time  for  solution  occupies  from  seven  to  twenty  hours. 

FIRST  GROUP  SECOND  GROUP  THIRD  GROUP 

Use  of  charts  Compass  German 

Computation  of  altitude  True  course  Algebra 

Longitude  and  time         Latitude  Planimetry 

Lunar  distances  Latitude  from  stars  Stereometry 

Position  by  two  altitudes  Position  by  dead  reckoning  Plane  trigonometry 

Variation  of  compass       Winds  and  currents  Spherical  trigonometry 

Signals  Log  book  Physics 


143  j     The  Present  Teaching  of  Mathematics  in  Germany      77 

In  the  first  group  all  problems  must  be  correct;  but  four  out 
of  each  of  the  other  groups  are  sufficient. 

The  amount  of  pure  mathematics  needed  is  merely  algebra, 
geometry,  and  trigonometry;  but  students  entering  a  school  of 
navigation  have  been  out  of  school  so  long  that  they  require 
much  review  of  elementary  principles.  J'hey  have  also  devel- 
oped habits  of  concrete  thought  which  make  abstract  concep- 
tions hard  for  them  to  grasp;  therefore  the  work  is  limited 
to  mere  preparation  for  the  examinations.  Then,  too,  the 
examinations  themselves  have  followed  a  stereotyped  plan  so 
that  the  candidate  knows  just  what  sort  of  problems  to  expect. 
This  also  prevents  any  broadening  of  methods  in  the  mathematics 
work.  Some  changes  in  the  requirements  or  methods  of  the 
examinations  might  well  be  instituted  in  order  to  give  a  broader 
training.  The  requests  for  post-graduate  courses,  made  by  many 
who  feel  their  deficiencies,  and  the  training-ships  of  the  North 
German  Lloyd  are  influences  in  the  right  direction. 

The  method  of  selecting  teachers  also  needs  to  be  changed. 
At  present  there  are  two  classes  of  teachers, — those  who  have 
much  nautical  experience  but  little  education,  and  those  who 
have  a  thorough  education  but  little  nautical  experience.  A 
course  should  be  arranged  so  that  a  prospective  teacher  may 
acquire  both  education  and  experience.  Then,  too,  the  system 
of  appointment  and  promotion  is  such  that  the  teachers  in  these 
higher  schools  of  navigation  are  more  than  sixty  years  old.  This 
gives  an  undue  conservatism  to  their  influence. 

The  system  of  examinations  has  been  very  beneficial  in  bring- 
ing seamanship  up  to  a  high  standard,  and  the  schools  of  navi- 
gation have  done  well  in  handling  the  material  which  comes  to 
them ;  but  many  changes  will  soon  be  necessary.  Modern  inven- 
tions are  making  navigation  into  a  new  science  and  removing 
the  need  for  many  of  the  former  methods.  The  schools  should 
keep  abreast  of  this  progress. 


CHAPTER  XII 

COMMERCIAL  PROBLEMS  IN  THE  HIGHER  SCHOOLS 
OF  GERMANY 

W.  F.  Enteman 

In  this  review  of  a  report  on  the  "  Commercial  Problems  in 
the  Mathematical  Instruction  of  the  Higher  Schools,"1  will  be 
found,  as  in  the  original  publication,  (a)  some  discussion  of  the 
field  of  commercial  problems,  (b)  a  brief  review  of  text-books, 
both  old  and  new,  (c)  a  discussion  of  the  important  aids  in 
teaching  the  subject,  and  (d)  some  general  conclusions.  One 
of  the  aims  of  the  report  is  to  call  attention  to  the  cause  for 
the  existence  of  commercial  problems  in  the  mathematical  in- 
struction in  Germany. 

The  German  Minister  of  Education  has  said,  "  Not  book 
knowledge  but  understanding  of  life  is  desired."  Shall  the 
schools  tackle  the  question?  Those  who  believe  the  object  of 
schools  is  the  education  of  young  men  for  citizenship  answer  in 
the  affirmative ;  those  who  believe  the  school  is  for  the  intellectual 
only  answer  in  the  negative.  In  the  opinion  of  Dr.  Timerding 
commercial  problems  have  a  meaning  from  either  standpoint, 
and  in  support  of  this  view  he  makes  the  following  statements. 

(1)  The  calculations  of  practical   problems   have  the   same 
psychological  value  as  the  calculations  of  long  tedious  problems 
that  have  no  bearing  on  life. 

(2)  Proofs  for  the  mathematical  formulas  needed  develop  the 
formal  side. 

(3)  Mathematics  is  as  valuable  considered  as  a  useful  tool  as 
when  it  is  looked  upon  merely  as  an  intellectual  product. 

(4)  Satisfaction  is  derived  from  the  knowledge  of  the  adapta- 
bility of  abstract  formulas. 


*Die    Kaufmannischen    Aufgaben    im    Mathematischen    Unterricht    der 
Hoheren  Schulen,  von  Dr.  H.  E.  Timerding,  Leipzig  und  Berlin,  1911. 
78  [144 


145]     The  Present  Teaching  of  Mathematics  in  Germany      79 

There  is  no  definite  standard  imposed  on  the  text-books  as  to 
the  amount  of  work  to  be  devoted  to  commercial  problems 
or  as  to  the  different  topics  to  be  treated.  The  personality  of 
the  teacher  is  the  important  factor  in  determining  this  amount. 
He  can  give  flesh  and  blood  to  the  framework  by  securing  local 
problems  of  vital  interest,  or  he  can  make  flimsy  the  work  of 
the  best  text-book.  In  arranging  a  course  of  study,  a  scientific 
method  should  be  followed  and  not  a  whimsical  one.  It  is  not 
a  simple  matter  to  correlate  the  theory  and  practical  applications 
of  any  science  in  teaching,  and  this  is  especially  true  with  regard 
to  mathematical  instruction.  Commercial  arithmetic,  for  exam- 
ple, must  be  treated  with  theoretical  arithmetic  as  heretofore, 
and  as  a  separate  subject,  but  a  far  more  definite  method  must 
be  worked  out. 

The  condition  of  the  work  rather  than  the  manner  of  instruc- 
tion is  the  main  topic  considered  in  this  report  and  the  following 
points  are  emphasized: 

(1)  That  it  is  possible  to  make  a  collection  of  scientific  prob- 
lems based  upon  practical  knowledge. 

(2)  That  the  empty  formalism  of  problems  must  be  discour- 
aged. 

REVIEW  OF  THE  FIELD  OF  COMMERCIAL  PROBLEMS 

For  a  general  view  of  the  field  from  which  problems  may 
be  drawn  we  must  consider  the  forms  of  our  commercial  activity 
in  general.  There  are  two  of  these  general  forms:  (i)  Traffic 
in  Goods,  and  (2)  Traffic  in  Money.  The  tradesman  is  espe- 
cially concerned  with  the  first,  the  banker  with  the  second.  The 
tradesman  must  be  able  to  find  a  suitable  sale  price  for  an  article 
when  the  cost  price  is  given,  or,  if  he  has  a  fixed  sale  price,  he 
must  find  a  suitable  cost  price  in  order  that  he  may  make  a 
certain  profit.  The  subject  of  alligation  is  applicable  to  problems 
in  this  first  field.  It  is  taught  in  the  general  schools,  and  usually 
in  connection  with  linear  equations.  It  treats  of  the  proportions 
in  which  goods  can  be  mixed  for  pleasant  taste  or  for  cheap- 
ness, as,  for  instance,  different  grades  of  tea,  coffee,  wine,  etc. 
The  fundamental  principle  is  simple;  namely,  that  the  total  cost 


8o  Teachers  College  Record  [146 

of  a  quantity  of  mixture  must  equal  the  cost  of  its  separate  con- 
stituents.   The  formula 

m1  pt  +  m2  p2  +  m8  p3  +.     .     .     . 


mt  +  m2  4-  ms  +.     .     .     . 

in  which  mt,  m2,  m3,  ....  stand  for  amounts,  and  plf  p2, 
p3,  .  .  .  .  stand  for  price,  can  be  used  if  the  price  of  a 
mixture  is  to  be  found  when  the  price  and  amounts  of  its  con- 
stituents are  given;  or  if  there  are  only  two  constituents  to  be 
used,  the  quantity  of  one  of  them  being  given  to  find  the  quan- 
tity of  other  that  will  give  a  mixture  worth  a  certain  price.  For 
an  illustration  see  Feller  and  Odermann,  "  Das  Ganze  der  kauf- 
mannischen  Arithmetik,"  Leipzig,  1908.  Another  use  for  alliga- 
tion is  in  the  solution  of  problems  on  the  coinage  of  money.  It 
may  be  used  to  find  the  value  of  a  known  mixture  of  metal  of 
which  certain  coins  are  composed,  or  the  weight  of  a  certain  kind 
of  precious  metal  which  is  to  be  mixed  with  a  known  weight  of 
another  metal  to  obtain  a  required  mixture.  Also  it  can  be  used 
to  find  the  proportion  of  alloy  and  metal  for  the  standard  coins, 
or  to  find  the  real  value  of  money  when  the  constituent  parts 
are  given.  This  is  treated  rather  superficially  in  the  general 
schools.1 

The  calculations  which  have  to  do  with  money  are  in  general 
of  greater  worth  and  interest  to  the  schools  than  any  others. 
We  can  divide  this  subject  into  two  parts  :  Primary  Calculation, 
as  in  the  case  of  simple  interest;  Higher  Calculation,  as  in  the 
case  of  compound  interest  and  the  more  advanced  work  in  simple 
interest. 

Under  Primary  Calculation  may  be  placed: 

(a)  Percentage,  with  the  important  formula  (i)  p  =  br. 

(b)  Interest,  the  fundamental  formula  of  which  is   (2)   1  = 
prt. 

For  convenience  in  account  current  the  interest  rate  is  reduced 
to  a  rate  per  day.  If  tlf  t,,  tg  .  .  .  indicates  the  time  in  days, 
and  alf  a2,  as  .....  the  deposits  which  lie  in  the  bank  up 


1  The  extensive  use  of  alligation  in  Germany  is  one  of  the  surprises 
that  greet  the  American  student.  There  is  at  present  a  little  effort 
to  revive  the  study  in  the  United  States,  but  its  value  seems  very  slight 
and  the  chance  of  its  introduction  in  the  schools  is  rather  remote. 


147]     The  Present  Teaching  of  Mathematics  in  Germany      81 

to  the  time  when  the  account  is  closed,  we  have  the  following 
formula  : 

(3)  I=(a1t1  +  a2t,  +  a3t3+    .......  )q,  in  which  q  = 

—  .    This  method  of  calculating  interest  is  called  the  progressive 

method.    If  the  accounts  are  closed  every  half  year,  the  follow- 
ing formula  is  used  : 

(4)  1=  [(a1  +  az  +  a,  .  .  .  .)i8o  —  a^iSo  —  tt)  —  a2(i8o- 


The  method  of  calculating  interest  by  this  formula  is  known 
as  the  retrograde  method.  Another  formula  often  used  in  de- 
termining interest  is: 

(5)  I=[a1(t1-tt)  +  (a1  +  af)(t±-t.)  +  (a1  + 


(c)  Discount. 

(6)  A  =  a(i  +  rt).     This   is  called  the  Hoffmann   discount 
formula  (1731). 

(7)  a'  =  A    (i  — rt).     This    is   called   the   Carpzo   discount 
formula  (1734).    These  two  correspond  to  our  formulas  for  true 
discount  and  bank  discount.     In  7,  a'  is  an  approximate  value 
for  a  when  rt  is  a  small  fraction. 

(d)  Equation  of  Payments. 

at      -4—    Q     t*       —I—    3     f"      —I— 
1  Lf         |        el.,  L.>        ]        do  Lo        ]  *          •          •          • 

A     L  mm  o    o 

(8)  t=r~rTT 

aj  -f  a2  ~r  a3  T- 

(e)  Partnership.     This  has  to  deal  with  the  distribution  of  a 
certain  profit  among  a  given  number  of  partners  according  to 
a  given  ratio.    If  A  represents  the  amount  that  is  to  be  divided, 

Ci,c2ic5 the    given    ratio,  and  A1,A2,A3   ....  the 

amount  that  each  should  receive,  we  have  the  formula 

CjA  c2A 

ct  +  c2  -f-  .    .    .  c,  +  c2  +  .    .    .    . 

Under  Higher  Calculation  may  be  placed: 
(a)  Compound  interest,  the  fundamental  formula  of  which  is 

(10)  A  =  p(i  +r)*. 

The  next  formula  is  especially  interesting  when  n  becomes 
indefinitely  large  and  the  interest  is  constantly  due  and  constantly 
added  to  the  capital.  Here  we  may  use  instead  of 

(11)  A- 


82  Teachers  College  Record  [148 

If  this  is  compared  with  (10)  we  have 

r'  =  nat  log  (i  +  r), 
which  may  be  developed  into  the  series 

r       T*      T*    . 

r'  =  r    I--  + h ]. 

234 
This  last  formula  was  introduced  by  Jacob  Bernoulli  in  1690. 

(b)  Calculation  of  Income  or  Revenue.  The  payment  of  a 
debt  through  annual  payments  and  the  fixing  of  allotment  for 
the  payment  of  a  loan  may  be  put  under  this  subject,  and  the 
following  formula  for  solving  such  problems  is  given. 

(12)  S  =  s(i+q  +  q2  +  q84J, +qt-1), 

where  q  =  i  +  r.  This  formula  is  usually  written  as  follows : 


which  is  a  familiar  formula  in  geometric  progression. 

From  the  calculation  of  final  S  of  an  income  follows  directly 
the  calculation  of  the  cash  value  B,  we  have 


q*  (q  - 1) 

We  can  also  find  the  natural  earnings  or  increment  if  we  assume, 
instead  of  separate  income  payments  which  follow  at  regular 
intervals,  a  continuous  payment.  There  falls  by  this  process 
upon  the  time  element  dt  an  infinitely  small  value  8  dt  and  the 
cash  value  of  income  will,  if  payment  is  for  the  period  t0  to  tlf 
be  expressed  by  the  integral 


(15)  B= 

J   • 

Instead  of  a  constant  stream  of  income,  a  variable  one  may  be 
used  in  which  S  is  a  function  of  the  time  and  we  must  write 
S(t).  Then  we  have 


(16)  B=     e-^t)  dt 
(c)  Insurance. 

REVIEW  OF  TEXT-BOOKS     . 

It  is  a  great  task  to  review  all  the  works  on  commercial  arith- 
metic, since  every  arithmetic  takes  up  at  least  something  on  the 
subject.  Only  a  few  of  the  most  important  works  can  be  men- 
tioned at  this  time. 


149]     The  Present  Teaching  of  Mathematics  in  Germany      83 

Martus  (1903)  "Mathematical  Problems  for  the  Upper 
Classes."  Besides  interest  and  partnership  problems,  there  are 
other  problems  which  deal  with  time  in  relation  to  money  cal- 
culation. As  an  illustration  of  a  problem  with  an  incorrect 
result  the  following  may  be  given  from  this  book : 

How  much  must  a  man  30  years  of  age  deposit  in  a  bank 
annually  until  he  is  65  years  of  age  in  order  that  he  may  receive 
3,000  Marks  yearly  for  10  years,  if  3l/2%  compound  interest  is 
allowed?  The  result  given  is  393  Marks,  but  no  consideration 
is  given  to  the  fact  that  the  man  may  die  before  he  is  75  years 
of  age.  The  correct  result  is  162  Marks.  "  Why  permit  a 
problem  with  an  incorrect  result  ?  "  is  the  comment  of  the  author. 

Bardy's  "Arithmetic"  (new  edition,  1910).  This  gives  real 
problems  only  in  compound  interest  and  calculations  on  income, 
and  rather  more  work  of  this  kind  than  necessary. 

Miiller  and  Kutnewsky  have  a  book  with  complete  tables  in 
insurance.  This  enables  the  teacher  to  choose  problems  to  suit 
himself  and  the  needs  of  the  class. 

A.  Schiilke  (1906)  has  a  text  that  covers  many  topics.  The 
applications  of  the  commercial  problems  are  very  accurate 
throughout.  Life  insurance  problems  are  given  and  a  general 
formula  is  stated,  but  not  a  single  example  under  this  formula 
is  given.  Lottery  problems  are  treated  very  fully,  a  feature  that 
is  practical  in  Europe  although  happily  obsolete  here.  In  spite 
of  some  defects,  this  seems  to  be  the  best  book  examined.  The 
problems  are  taken  from  real  life. 

Schulz  and  Pahl  (1906)  are  the  authors  of  a  book  which  is 
in  many  respects  the  opposite  of  the  one  just  reviewed.  Com- 
pound interest  and  investment  problems  are  very  few.  Mor- 
tality problems  and  savings  bank  problems  are  given,  but  no  life 
insurance  problems  appear. 

A  Bavarian  text-book  by  Hoffman  (1892)  has  much  that  is 
commendable.  Life  insurance  is  treated  briefly  but  skillfully. 

In  general  these  new  books  make  an  effort  to  obtain  problems 
from  the  experience  of  the  citizens  and  to  place  before  the 
student  statements  that  will  give  him  sound  and  rational  notions 
on  the  subject.  It  is  difficult  to  find  texts  to  supplement  the 
work  in  commercial  arithmetic.  Feller  and  Odermann  have  a 


84  Teachers  College  Record  [150 

book  that  is  well  written  but  it  is  not  one  that  can  be  given  to 
the  pupils. 

Thaer  and  Rouwolf  (1911)  treat  commercial  problems  in  an 
abbreviated  form. 

Moritz  Cantor's  "  Political  Arithmetic  "  is  a  small  book,  but 
a  very  good  one.  Dr.  Timerding  makes  the  following  comment 
on  this  work :  "  In  the  work  of  selecting  problems  a  word  of 
caution  may  be  necessary.  There  is  danger  that  this  attempt 
to  select  interesting  problems  for  the  student  may  be  carried 
too  far.  Cantor  is  whimsical,  being  inclined  to  skip  about,  and 
may  not  always  want  what  is  best." 

The  so-called  story  problems  fall  into  two  classes:  (i)  real; 
(2)  unreal  or  fanciful.  If  only  formal  training  is  desired,  the 
latter  class  of  problems  with  their  usual  rules  is  successful  with 
young  minds.  If,  on  the  other  hand,  immediate  practical  instruc- 
tion is  the  aim,  real  problems  are  necessary.  In  the  old  books 
the  unreal  problems  were  separated  from  the  others.  Some  of 
these  problems  which  appear  to-day  in  our  texts  are  of  great  age ; 
and  the  origin  of  many  can  be  traced  to  Egypt,  India,  Greece 
and  China.  The  following  is  an  example  of  an  old  Egyptian 
problem.  How  many  pigeons  are  on  a  ten-round  ladder,  if  there 
is  one  on  the  first  round,  two  on  the  second,  four  on  the  third, 
etc.?  The  charm  is  in  the  result  that  there  would  be  512  pigeons 
on  the  last  round. 

Some  of  the  fairy  problems  are  so  old  that  no  one  knows 
where  they  came  from.  The  following  was  taken  from  a  book 
which  appeared  in  1494:  On  the  top  of  a  tree,  which  is  60  ells 
high,  sits  a  mouse  and  on  the  ground  beneath  the  tree  is  a  cat. 
The  mouse  climbs  down  £  ell  each  day  and  climbs  up  £  ell  each 
night.  The  cat  climbs  up  i  ell  each  day  and  climbs  down  £  ell 
each  night.  The  tree  grows  ^  ell  each  day  and  shrinks  £  ell 
each  night.  When  will  the  cat  reach  the  mouse  and  how  high 
will  the  tree  be  at  that  time? 

In  the  works  of  Paciuolo  (1494)  we  first  find  interest  cal- 
culation.1 The  calculation  of  interest  by  means  of  the  linear 
equation  is  given  in  a  work  of  the  Renaissance  period,  and  in 
1600  appears  in  an  arithmetic.  Stevin  in  1582  gave  an  interest 

1  This  is  not  true,  for  such  work  appears  in  a  number  of  books  pub- 
lished before  1494. 


151]     The  Present  Teaching  of  Mathematics  in  Germany      85 

table ;  Jost  Biirgi  introduced  some  calculations  with  decimal  frac- 
tions j1  Insurance  appeared  during  the  seventeenth  century ;  Hal- 
ley,  in  1693,  for  the  first  time  found  the  present  worth  of  an 
annuity  based  upon  mortality  tables.  The  leading  features  of 
Commercial  Arithmetic  began  to  develop  in  the  eighteenth  cen- 
tury. Halcke,  in  1719,  published  his  "  Sweet  Meat  Thoughts," 
much  of  the  work  being  in  verse.  Many  works  appeared  at  this 
time,  and  among  the  number  may  be  mentioned  those  of  Claus- 
berg  (1732)  and  Biisch  (1769). 

About  the  beginning  of  the  nineteenth  century  the  subject  was 
taken  up  in  the  higher  schools  and  the  cultural  side  was  empha- 
sized. Lately  the  idea  of  money  making  has  brought  the 
practical  side  forward.  The  formal  side  has  much  worth,  the 
advantage  of  which  may  be  seen  if  we  view  it  from  problems 
which  are  not  merely  fanciful.  It  seems  wrong  to  propose  ques- 
tions which  in  practice  are  meaningless.  The  writer  of  this 
report  says  that  he  is  not  opposed  to  the  formal  character  of 
problems  if  the  aim  of  each  problem  can  be  made  clear.  A  good 
example  of  the  above  type  is  one  from  Euler  which  results  in  an 
equation  of  the  third  degree  with  three  positive,  integral  roots, 
namely  7,  8  and  10.  We  can  understand  the  problem  better  if  we 
write  the  equation  in  the  following  form: 

(x-7)  (x-8)  (x-io)-o, 

y^ 

or    (8240  +  4Ox2)  — —  i  ox2  =  224. 
100 

The  problem  given  is  this :  There  were  x  persons  engaged  in 
a  certain  business  and  to  their  joint  capital  of  8,240  Marks  each 
added  4Ox  Marks.  Their  profits  amounted  to  x  per  cent,  and  of 
this  profit  each  took  iox  Marks  and  then  there  remained  224 
Marks.  It  is  required  to  find  x.  The  interest  in  this  problem 
lies  in  the  fact  that  there  are  three  solutions. 

Another  good  example,  if  considered  from  the  formula  stand- 
point, is  the  following:  Three  brothers  inherit  45,500  Marks. 
The  one  that  received  the  least  amount  gave  -fa,  the  one  that  re- 
ceived the  most  gave  £,  the  third  gave  £  to  a  charitable  institu- 
tion, after  which  they  all  had  the  same.  How  much  did  each 


'This  is  a  superficial  statement,  not  to  be  taken  seriously. 


86  Teachers  College  Record  [152 

receive  ?  The  problem  sets  forth  a  generosity  not  found  in  com- 
mon life. 

The  following  problem  from  Schiilke  is  an  excellent  one: 
Two  persons  take  out  life  insurance.  Each  pays  100  Marks 
yearly,  one  for  30  years  and  the  other  for  10  years.  Would  it 
be  the  same  for  the  company  if  each  paid  for  20  years? 

In  compound  interest,  neither  the  time  nor  the  rate  is  ever 
unknown,  nor  is  it  practical  to  count  the  interest  for  more  than 
10  years  or  for  such  rates  as  3$%  or 


AIDS 

The  principal  thing  to  be  emphasized  in  the  treatment  of 
commercial  problems  is  simple  numerical  calculation.  Calcula- 
tion is  the  backbone  of  the  arithmetic  course  of  study  just  as 
proof  is  that  of  geometry.  The  defective  preparation  of  pupils 
in  the  higher  schools  in  practical  calculation  is  a  decided  fault. 
Some  of  the  things  which  might  be  mentioned  as  an  aid  in  the 
presentation  of  commercial  problems  are  : 

(1)  Graphic  representation.    Often  the  meaning  of  a  problem 
as  well  as  the  meaning  of  the  result  can  be  made  clear  by  a 
drawing.     Some  of  the  means  of  representation  are  very  old. 
From  the  works  of  Leonardo  of  Pisa  we  find  a  problem  involv- 
ing proportion,  the  solution  of  which  is  represented  by  lines. 

(2)  Slide  rule.     Next  to  graphic  aids  the  slide   rule  is  of 
greatest  value  to  the  engineer.    It  is  not  generally  used  in  con- 
nection with  commercial  problems. 

(3)  Logarithmic  tables  and  logarithmic  curves. 

(4)  Formula.    The  formula  is  perhaps  rated  too  high.     It  is 
not  of  so  much  importance  in  commercial  as  in  technical  arith- 
metic. 

(5)  Simple  illustrations  to  explain  the  solution  of  a  problem. 
The  following  taken  from  Findeisen-Clausen's  "  Examples  and 
Problems  for  Instruction  in  Commercial  Calculations  "  (Leipzig, 
1905)  will  explain  the  above.     If  we  have  a  problem  in  which 
we  are  to  find  the  amount  of  a  note  due  at  some  future  time 
and  also  to  change  from  one  standard  to  another,  we  might  use 
a  river  as  an  illustration.    The  two  banks  will  correspond  to  the 


The  Present  Teaching  of  Mathematics  in  Germany      87 

two  standards.  The  flowing  of  the  water  will  correspond  to  the 
passing  of  the  time.  The  crossing  of  the  river  corresponds  to 
the  changing  from  one  standard  to  the  other.  As  there  are  two 
ways  of  going  from  a  point  on  one  side  of  the  river  to  a  point 
farther  down  the  river  and  on  the  opposite  side,  so  there  are  two 
ways  of  solving  the  given  problem.  A  person  can  go  directly 
across  the  river  and  then  go  to  the  other  point  or  he  may  go 
down  the  river  to  a  point  directly  opposite  the  other  point  and 
then  cross.  Likewise  in  solving  the  problem,  we  may  change 
from  one  standard  to  the  other  and  then  calculate  the  amount, 
or  we  may  calculate  the  amount  first  and  then  change  from  the 
given  standard  to  the  other  standard.  Also  the  formulas  re- 
lating to  mixtures  and  alloys  are  analogous  to  the  one  relating 
to  the  center  of  gravity.  The  separate  articles  correspond  to 
the  different  points,  and  the  different  prices  to  the  different  dis- 
tances. Equation  of  payments  could  also  be  used  with  the 
formula  for  the  center  of  gravity.  The  different  amounts  cor- 
respond to  the  different  points,  and  the  periods  of  time  to  the 
distances. 

(6)  Graphs.  The  graphic  method  leads  only  to  an  approxi- 
mate result,  but  solutions  involving  several  decimal  places 
have  no  place  in  reality.  The  graph  has  already  won  recogni- 
tion in  the  schools,  and  if  millimeter  paper  is  used,  sufficiently 
accurate  results  can  be  obtained.  We  might  distinguish  two  dif- 
ferent methods. 

(a)  Graphic  statistics.     In  this  case  we  construct  a  curve  to 
show  at  a  glance  the  general  inference  to  be  drawn  from  a  set  of 
statistics.    As  an  illustration  we  could  have  a  curve  showing  the 
number  of  people  out  of  1,000,  say,  that  die  at  the  different 
ages.     This  curve  starts  at  a  considerable  distance  from  the 
abscissa  and  very  irregularly  approaches  the  abscissa  which  it 
finally  reaches. 

(b)  Graphic  calculation.  If  we  have  given  a  curve  constructed 
from  statistics,  as  in  the  example  under  (a),  or  from  a  tabula- 
tion of  the  results  of  experiments,  we  can  obtain  the  value  for 
any  abscissa  directly  from  the  curve  without  referring  back  to 
the  tables,  and  can  easily  interpolate  values  not  given  in  the 


88  Teachers  College  Record  [154 

tables.  A  graphic  table  may  be  used  in  connection  with  arbi- 
trated exchange.  It  may  be  shown  easily  and  quickly  whether 
it  would  be  better  to  send  directly  to  another  city  or  to  send 
through  a  third  place. 

CONCLUSION 

We  have  seen  that  there  are  two  opposing  views  in  regard 
to  the  study  of  commercial  subjects.  The  advocates  of  the 
one  view  demand  the  consideration  of  practical  claims  and 
simple  correctness  of  the  problems,  with  everything  in  harmony 
with  actual  business  transactions.  The  others  claim  that  it  is 
wrong  to  emphasize  excessively  business  practice,  as  against 
theory,  and  to  permit  this  to  interfere  with  a  scientific  treatment 
of  the  subject. 

We  should  not  endeavor  to  make  the  pupil  an  accomplished 
merchant;  he  should  only  be  expected  to  acquire  some  knowl- 
edge of  the  fundamental  principles  of  commercial  transactions. 
It  is  not  necessary  that  he  should  know  all  of  the  technical  terms 
and  all  of  the  finer  details  connected  with  commercial  calcula- 
tions. We  should  seek  to  impart  only  a  sound  practical  knowl- 
edge through  simple  problems  which  are  suitable  from  the  stand- 
point of  the  pupil. 

The  real  problems  should  be  separated  from  the  unreal  so 
that  the  pupil  may  know  which  kind  he  is  working  with.  The 
aim  of  unreal  problems  is  to  arouse  the  play  instinct  of  the 
young  mind  and  thus  to  animate  his  study  of  mathematics  and 
in  this  they  play  an  important  part.  The  result  at  which  they 
aim  is  formal.  The  pupil  learns  from  amusing  and  interest- 
ing problems  a  new  mathematical  process  or  finds  a  new  ap- 
plication for  a  process  of  calculation  already  learned.  On  the 
other  hand,  the  real  problem  leads  him  to  a  simple  problem  full 
of  meaning.  It  gives  him  a  moment  of  real  life.  It  shows  him 
how  the  things  he  learns  in  school  can  be  applied  to  things  out- 
side. It  is  placing  this  directly  before  him  that  binds  school 
and  life  together. 

From  this  report,  we  might  draw  the  following  conclusions: 

(i)  There  is  no  definite  course  of  study  in  commercial  arith- 
metic in  the  higher  schools  of  Germany. 


155]     The  Present  Teaching  of  Mathematics  in  Germany      89 

(2)  The  subject  of  alligation  is  considered  to  be  important, 
although  in  this  report  only  the  simpler  cases  were  mentioned. 

(3)  The  formula  is  used  in  the  solution  of  problems  to  a 
much  greater  extent  than  is  done  in  this  country. 

(4)  The  theory  underlying  certain  subjects,  as  life  insurance, 
theory  of  probabilities,  etc.,  is  considered  much  more  fully  than 
in  this  country. 

(5)  The  graph  is  used  to  a  great  extent. 

(6)  The  question  of  real  problems  is  with  them  an  unsettled 
one  just  as  it  is  with  us.     On  the  whole,  it  may  be  said  that 
the  work  that  we  in  the  United  States  are  doing  in  commercial 
arithmetic  compares  very  favorably  with  the  work  being  done 
at  present  in  Germany.     At  any  rate,  we  seem  to  have  fewer 
problems  that  are  unreal  while  pretending  to  represent  modern 
conditions. 


CHAPTER  XIII 
MATHEMATICS   IN   THE  TEXT-BOOKS   ON   PHYSICS 

Arthur  T.  French 

This  article  contains  a  brief  digest  of  a  much  more  extended 
discussion  under  the  same  title1  in  one  of  the  reports  of  the 
International  Mathematics  Commission. 

Although  Dr.  Timerding  has  written  in  the  interests  of  mathe- 
matical instruction,  it  has  not  been  his  intention  to  emphasize 
mathematics  unduly  or  to  prescribe  rules  for  the  teaching  of 
physics.  He  does,  however,  aim  to  show  that  mathematical 
problems  arise  in  the  study  of  physics  and  that  the  solution  of 
these  problems  is  necessary  for  an  understanding  of  the  physics. 
Many  of  the  difficulties  of  physics  will  be  made  easy  through 
a  reform  in  the  teaching  of  mathematics. 

Physics  instruction  ought  to  be  concrete  and  pupils  should 
perform  their  own  experiments.  In  the  lower  grades  this  in- 
struction should  be  given  without  any  mathematics.  The  in- 
troduction of  mathematics  too  early  in  the  grade  lessens  the 
interest  in  the  subject;  but  before  beginning  the  formal  study 
of  physics  the  pupil  should  have  a  fair  mastery  of  all  neces- 
sary mathematics.  How  much  that  includes  is  a  difficult  ques- 
tion to  answer,  but  in  the  beginning,  mathematics  and  physics 
should  be  independent.  In  the  majority  of  schools,  instruc- 
tion in  mathematics  and  instruction  in  physics  go  side  by 
side,  consequently  the  pupil  cannot  use  his  knowledge  of 
the  one  subject  in  the  study  of  the  other.  This  arrangement 
is  not  necessary  since  one  can  just  as  well  be  given  be- 
fore the  other.  Mathematics  ought  to  furnish  exact  concepts 
to  physics  and  physics  ought  to  furnish  problems  for  mathe- 
matics. Thus  both  subjects  would  be  enlivened  and  strengthened. 

1  Die  Mathematik  in  den  Physikalischen  Lehrbiichern,  von  Dr.  H.  E. 
Timerding.  O.  Professor  an  der  Technischen  Hochschule  in  Braunschweig. 
Leipzig  and  Berlin,  1910. 

90  [156 


I57]     The  Present  Teaching  of  Mathematics  in  Germany      91 

It  is  not  necessary  to  weaken  mathematics  or  to  lose  the  ob- 
jectivity of  physics. 

The  physics  teacher  of  to-day  has  a  difficult  task,  because  he 
cannot  take  for  granted  that  his  pupils  possess  a  definite  amount 
of  knowledge  of  mathematics.  This  is  especially  true  in  the 
university  where  the  amount  they  do  have  varies  so  much  that 
it  is  hard  to  adapt  the  instruction  to  the  needs  of  all.  He  cannot 
cater  to  the  well-prepared  but  must  adapt  his  instruction  to  fit 
the  needs  of  the  majority.  A  reform  has  already  begun  in  the 
university,  but  it  has  not  yet  affected  the  lower  classes  to  any 
extent. 

The  discussion  which  follows  is  based  solely  upon  the  ex- 
amination of  text-books.  The  author  realizes  that  good  teach- 
ing is  as  essential  as  a  good  text,  but  an  investigation  of  the  work 
of  teachers  would  necessarily  be  very  limited.  Reports  from 
teachers  upon  the  methods  used  would  probably  confirm  the 
statements  of  the  author.  This  examination  has  been  confined 
in  general  to  the  most  popular  German  texts,  the  exceptions 
being  in  the  case  of  those  that  are  remarkable  for  any  peculiari- 
ties, regardless  of  their  popularity. 

The  problem,  then,  as  the  author  sees  it,  is  to  examine  the 
material  at  hand  to  get  the  scope  and  character  of  that  part  of 
physics  which  is  usable  in  mathematics.  The  mathematics  of 
physics  is  distinctly  geometric,  as  physical  phenomena  take  place 
in  space.  To  get  the  scope  and  character  of  the  mathematics 
which  has  grown  from  physics,  as  it  is  at  present,  we  must  first 
consider  the  historical  development  of  physics. 

Physical  problems  came  first  and  the  mathematical  treatment 
had  to  be  sought ;  but  this  mathematical  development  was  prac- 
tically finished  when  physics  texts  began  to  be  written.  In  this 
development  has  physics  had  any  influence  on  modern  mathe- 
matical methods,  or  has  mathematics  vitally  influenced  physics 
or  physics  instruction?  What  was  the  origin  of  present  mathe- 
matical methods  in  physics?  Where  did  they  come  from,  and 
what  was  physics  before  this  time?  These  are  some  of  the  ques- 
tions to  be  answered. 

The  development  of  the  physics  texts  along  mathematical  lines 
begins  with  the  seventeenth  century.  Physics  ceased  to  be  a 


92  Teachers  College  Record  [158 

branch  of  Aristotelian  philosophy  and  became  a  branch  of  nat- 
ural philosophy  with  Galileo.  The  physical  text-book  originated 
with  Descartes,  an  opponent  of  Galileo's  ideas.  One  of  the 
earliest  texts  was  published  in  1672  in  Amsterdam,  by  Rohault. 
Both  Rohault  and  Descartes  treated  the  subject  quite  fantastically. 

The  mathematics  of  those  days  was  only  a  methodical  dis- 
cipline, and  was  first  used  as  a  dominant  method  in  physics  by 
Newton.  Newton's  "  Mathematical  Principles  of  Natural  Phil- 
osophy "  was  due,  however,  in  a  considerable  measure,  to  Des- 
cartes. The  first  result  of  Newton's  influence  was  a  book  written 
in  1721  by  Gravesande,  "  Physices  Elementa  Mathematica  Ex- 
perimentis  Confirmata,"  published  in  Leiden.  Gravesande  in 
this  work  states  that  physics  belongs  to  mathematics.  In  1725, 
at  London,  Desaguliers  published  his  "  Course  of  Experiments 
in  Philosophy  "  as  a  result  of  Newton's  influence  and  lectures. 
Desaguliers  declared  that  physics  without  observation  and  ex- 
periment is  useless,  but  that  we  must  bring  geometry  and  arith- 
metic to  our  aid  unless  we  wish  these  observations  and 
experiments  to  be  pure  guesswork.  To  become  a  physicist  one 
must  know  mathematics,  although  Newton's  principles  can  be 
imparted  to  the  general  public  without  it.  This  is  in  accord 
with  the  ideas  of  modern  teachers. 

In  Segner's  "  Introduction  to  Natural  Philosophy,"  published 
at  Gottingen  in  1746,  stress  is  laid  on  geometry,  which  the  author 
declares  to  be  the  foundation  of  all  natural  science,  although 
the  more  difficult  and  less  known  principles  of  the  subject  may 
be  avoided.  Arithmetic  may  sometimes  be  used  in  place  of 
geometry. 

The  mathematical  ideal  pervaded  the  natural  philosophy  of 
the  eighteenth  century.  A  certain  group  of  scholars,  however, 
worked  along  quite  different  lines  from  the  rest  and  assumed 
an  attitude  of  dilettanteism.  The  work  of  this  group  was  signifi- 
cant, as  it  paralleled  the  dawn  of  the  modern  conception  of 
electricity  and  magnetism  which  completely  revolutionized  the 
methods  of  physical  instruction.  Thus  we  find  that  after  a  long 
dominion  of  the  mathematical  method,  the  inductive  experimental 
method  came  into  the  foreground.  The  theory  of  electricity  was 
not  taken  up  in  texts  until  its  susceptibility  to  mathematical  treat- 


I59l     The  Present  Teaching  of  Mathematics  in  Germany      93 

ment  seemed  certain;  but  professional  men  were  inclined  to 
regard  it  as  play  rather  than  an  earnest  science. 

The  first  part  of  the  nineteenth  century  saw  many  new  texts. 
Gottfried  Fischer's  "  Lehrbuch  der  Mechanischen  Naturlehre," 
published  in  1805,  is  the  basis  for  many  of  our  modern  texts. 
It  was  not  used  to  any  extent  in  Germany,  but  it  was  translated 
into  French  and  became  a  very  popular  text  with  the  French 
people. 

It  is  interesting  to  consider  the  origin,  growth,  and  methods 
of  the  text-books  in  physics.  A  text-book  is  the  outgrowth  of 
a  definite  need  in  teaching,  and  this  need  is  especially  great  in 
physics.  As  the  teaching  of  physics  begins  in  the  lower  grades 
and  is  carried  into  the  university,  the  number  of  texts  on  this 
subject  increases  with  little  improvement  in  quality. 

Certain  characteristics  are  common  to  all, — conservatism, 
poor  diagrams  and  pictures,  and  lack  of  graphic  work. 
Conservatism  arises  from  the  fact  that  many  teachers  aim  to 
teach  the  subject  just  as  it  was  taught  to  them.  They  try  to 
adapt  new  material  to  old  methods,  and  prefer  a  text  that  clings 
to  the  old  traditions  and  resembles  as  nearly  as  possible  the  one 
they  used.  The  teacher  also  prefers  a  text  that  makes  the  in- 
struction as  easy  as  possible.  This  is  especially  true  in  the 
university,  where  the  lecture  method  is  used  and  where  the  aver- 
age student  may  not  be  helped  at  home.  For  this  reason,  a  text 
is  apt  to  have  a  long  lease  of  life.  The  Miiller-Pouillet  text  is 
a  good  illustration.  It  is  sixty  years  old,  and  is  taken  from 
another  eighty  years  old,  although  it  has  been  revised.  Those 
books  that  are  not  original  are  most  successful ;  a  writer  cannot 
afford  to  introduce  too  many  novelties.  Mach's  book  is  a  fine 
original  text,  but  it  has  not  been  a  success  largely  for  this  reason. 

Another  reason  for  the  obstinate  holding  to  the  old  ideas  is 
that  the  amount  of  matter  to  be  treated  is  so  enormous  that  a 
text  cannot  be  written  from  memory,  so  that  the  writers  must 
depend  on  former  texts  as  a  source  of  material  and  as  a  model. 
Many  texts  have  been  written  in  this  way  without  much  thought 
being  put  upon  the  work.  However,  the  present  tendency  seems 
to  indicate  a  change  for  the  better. 

Another  fault  common  to  physics  texts  is  their  treatment  of 


94  Teachers  College  Record  [160 

the  problems  involving  the  calculus.  The  introduction  of  the 
conceptions  of  velocity  marks  the  critical  point  in  the  text-book, 
because  it  involves  the  idea  of  the  infinitesimal.  Differential 
calculus  is  the  only  sound  basis  for  a  conception  of  velocity, 
although  this  concept  may  be  presented  without  its  aid.  In 
fact,  this  concept  may  be  easily  given  to  the  ordinary  student 
of  physics.  One  may  conceive  of  non-uniform  velocity,  but  it 
cannot  be  fully  comprehended  without  the  idea  of  limits.  In 
the  newer  books  which,  on  principle,  avoid  any  such  introduc- 
tion, the  laws  of  gravity  are  presented  in  the  form  of  an  em- 
pirical formula  obtained  by  Atwood's  machine;  Cruger,  for 
example,  does  this.  Mach  gives  an  abridged  empirical  table 
which  is  worked  out  in  detail  in  his  latest  book. 

Inaccuracy  in  drawings  and  illustrations  is  another  char- 
acteristic of  the  texts.  These  are  of  great  importance  mathe- 
matically, as  may  be  shown  by  a  consideration  of  the  kinds 
usually  shown  in  texts.  The  first  thing  considered  by  the  text- 
book writer  is  the  expense,  and  this  has  led  to  the  use  of  many 
poor  drawings  and  many  illustrations  so  old  as  to  have  no  value 
to  the  pupil.  In  an  elementary  text  by  Koope-Husmann  there 
is  a  picture  of  a  rainbow  with  large  drops  of  water  visible. 
Gravesande  has  a  figure  of  a  steam  engine  and  of  other  machines 
over  two  hundred  years  old.  A  picture  of  a  primitive  type  of 
locomotive  was  in  use  for  over  fifty  years.  Only  in  the  very 
recent  books  are  there  any  really  new  drawings  or  pictures. 

One  method  for  improving  the  drawings  and  pictures  is  for 
a  publisher  to  have  some  very  good  cuts  and  use  the  same  in 
all  his  books.  This  is  done  by  a  house  in  Braunschweig  and  thus 
excellent  cuts  are  used  in  cheap  books.  One  objection  to  this 
plan  is  that  there  is  a  tendency  toward  their  use  for  ornamental 
purposes  in  which  case  their  significance  is  lost.  Mach  and 
Bremer  have  drawings  by  the  authors  which  are  very  poor. 
Kleiber-Karsten  have  many  drawings  to  bring  out  important 
facts,  but  they  are  of  inferior  quality.  There  are  advantages  in 
having  the  author  make  his  own  drawings  and  in  having  many 
of  them,  so  this  method  must  not  be  discarded  too  readily.  La 
Grange  boasted  of  never  having  used  a  figure  in  his  "  Elements 
of  Mechanics." 


161]     The  Present  Teaching  of  Mathematics  in  Germany      95 

For  anything  that  the  teacher  himself  cannot  show,  the 
pictures  ought  to  be  good;  for  example,  in  the  study  of 
large  machines  or  cloud  formation.  "  Schematische "  figures 
may  be  used  in  place  of  real  figures.  This  is  the  case  when 
the  mathematical  development  is  brought  in.  This  development 
should  not  be  carried  to  a  point  where  it  will  kill  the  living  con- 
ception of  the  thing  and  make  the  pupil  think  in  mathematical 
abstractions.  These  drawings  should  be  as  simple  as  they  can 
be  made.  Properly,  every  author  of  a  text-book  should  either 
be  a  good  draughtsman  or  should  collaborate  with  one.  It 
is  of  great  importance  that  every  drawing  shall  be  true  in  perspec- 
tive ;  otherwise  the  pupil  will  get  wrong  ideas  of  geometric  con- 
struction. For  example,  very  few  books  properly  represent  the 
sphere,  meridians  and  parallels  of  longitude.  Warburg  has  a 
false  perspective  drawing  of  a  cone  which  was  taken  from  a 
correct  drawing  in  Helmholtz.  If  the  pupil  does  not  have  exact 
pictures,  how  can  he  draw  correctly,  or  how  can  he  learn 
geometry  ? 

It  is  only  recently  that  any  importance  has  been  given  to  the 
kind  of  figures  and  pictures  in  a  text-book.  Correct  drawings 
have  a  great  educational  value,  because  the  student^  is  inspired 
by  the  idea  that  there  is  a  connection  between  all  branches  of 
human  knowledge,  and  correct  drawings  enable  him  to  get  some 
idea  of  this  connection.  For  example,  accurate  drawings  of 
rays  of  light  going  into  water,  of  waves  of  sound,  and  of  the 
refraction  of  rays  of  light  in  raindrops,  give  him  ideas  of 
geometric  curves. 

Graphic  representations  are  of  an  importance  only  recently 
recognized.  Without  exaggeration,  all  important  functional  phe- 
nomena that  occur  in  nature  may  be  pictured  by  curves.  The 
graphic  representation  of  different  functions  is  one  of  our  most 
important  means  of  teaching  mathematics,  therefore  physics 
aids  greatly  in  mathematical  instruction  since  various  physical 
phenomena  lend  themselves  to  graphic  treatment.  England  is 
largely  responsible  for  the  introduction  of  the  use  of  graphic  aid% 
in  instruction  in  physics.  In  Germany  to-day,  physics  and  graphic 
representations  are  so  far  apart  that  there  is  no  possibility  of 
applying  the  graph  to  physics,  although  it  was  used  in  a  treatise 


96  Teachers  College  Record  [162 

on  natural  philosophy  nearly  fifty  years  ago.  When  we  find 
any  use  of  the  graph  it  is  in  connection  with  thermodynamics. 
The  isothermic  lines  are  generally  used  to  give  an  idea  of  tem- 
perature and  of  the  critical  point.  Clausius  was  the  first  man  in 
Germany  to  do  this  kind  of  work  and  he  has  been  very  much 
interested  in  it. 

The  paper  just  summarized  is  interesting  for  one  thing, 
namely,  we  find  the  Germans  so  far  behind  us  in  the  matter  of 
texts.  For  some  time  the  necessity  for  good  drawings  and  for 
good  illustrations  in  texts  has  been  recognized  in  this  country, 
and  to-day  we  have  many  good  texts  which  are  very  satisfactory 
in  this  respect. 

We  have  in  this  country  two  bodies  of  physicists — those  inter- 
ested in  correlating  mathematics  and  physics,  and  those  interested 
in  keeping  them  as  far  apart  as  possible ;  but  the  college  exam- 
inations settle  quite  definitely  the  type  of  work  that  must  be 
done  by  schools  that  attempt  to  prepare  for  college,  so  most 
of  our  physics  is  the  so-called  mathematical  physics.  We  will 
agree,  I  think,  that  a  sufficient  amount  of  mathematics  should 
precede  physics,  but  as  to  how  much  that  should  be  we  are 
uncertain,  as  are  the  Germans.  The  ideas  of  the  calculus  may 
be  taught  much  earlier  than  we  teach  them,  and  to-day  ele- 
mentary calculus  is  being  given  in  some  high  schools.  In  gen- 
eral, Germany  seems  to  be  doing  as  little  as  we  are  to  correlate 
closely  the  mathematics  and  the  physics  of  the  schools. 


CHAPTER  XIV 

GOVERNMENT  EXAMINATIONS  IN  PRUSSIA  AND 
THE  NORTH  GERMAN  STATES 

Cilda  Langntt  Smith  and  Katherine  Simpson 

The  report  here  reviewed1  gives  the  important  regulations  that 
were  passed  in  regard  to  the  state  examinations  in  Prussia  and 
the  United  North  German  States  from  1810  to  1898,  and  also 
the  regulations  concerning  the  jexaminations  in  Braunschweig  and 
Mecklenberg.  Dr.  Lorey  shows  that  the  principal  reason  why 
Germany  has  accomplished  so  much  in  the  mathematical  field 
is  because  her  educators  early  conceived  the  idea  of  gradually 
increasing  the  mathematics  requirements  for  the  teachers,  and 
the  result  of  this  policy  is  that  the  schools  of  to-day  have  thor- 
oughly equipped  teachers  who  have  a  broader  knowledge  of  the 
subject  than  the  bare  contents  of  the  curriculum. 

The  higher  schools  of  Prussia  and  the  United  North  German 
States  were  established  many  years  before  1810.  These  higher 
schools  were  generally  founded  by  the  church  and  certain  com- 
missions. Their  teachers  were  the  clergymen  who,  while  waiting 
for  better  paid  positions,  regarded  this  as  the  most  profitable 
way  to  employ  their  time.  The  order  of  the  clergy  was  pre- 
ferred because  the  teaching  profession  at  that  time  was  con- 
sidered one  of  comparatively  low  rank. 

Two  minor  laws  in  regard  to  the  requirements  of  the  teachers 
were  passed  before  1810 ;  the  one  in  1718,  which  decreed  that  all 
the  teachers  in  the  German  and  Latin  schools  must  pass  an 
examination  before  a  committee,  the  other  in  1787,  which  decreed 
that  every  teacher  before  being  admitted  to  the  teaching  pro- 
fession must  hold  a  teacher's  certificate.  But  the  edict  of  1810 
was  the  first  one  which  was  really  enforced.  The  purpose  of 

1  Staatspriifung  und  Praktische  Ausbildung  der  Mathematiker  an  den 
Hoheren  Schulen  in  Preussen  und  Einige  Norddeutschen  Staaten,  von 
Dr.  Wilhelm  Lorey,  Pro-rektor  der  Kge.  Oberrealschule  in  Minden,  1911. 

163]  97 


98  Teachers  College  Record  [164 

this  edict  was  to  prevent  incapable  teachers  from  entering  the 
teaching  profession.  It  must  be  understood  that  the  states  in 
which  this  edict  was  to  take  effect  were  divided  into  three  de- 
partments of  instruction,  i.e.,  the  departments  of  Berlin,  Breslau, 
and  Konigsburg.  The  examinations  were  held  before  commit- 
tees chosen  by  these  departments.  The  following  teachers 
were  required  to  take  the  examinations:  future  teachers  of 
the  public  schools  who  were  preparing  pupils  for  the  second 
and  third  classes  of  the  Gymnasium  and  Realschule  (which 
classes  are  the  same  as  our  sixth  and  seventh  grades  respec- 
tively) ;  teachers  who  were  planning  to  teach  in  private  schools; 
and  teachers  of  the  public  schools  who  were  to  prepare  the  pupils 
for  entrance  to  universities.  Those  who  were  not  required  to 
take  the  examinations  were  teachers  in  the  elementary  schools 
(the  Folks-schulen  and  Biirger-schulen)  and  young  graduates 
of  the  universities,  who  were  planning  to  teach  only  for  a 
short  time.  One  can  scarcely  judge  the  standard  of  the  uni- 
versities at  this  time,  because  the  requirements,  as  outlined  in 
this  edict,  were  very  indefinite.  The  candidate  was  required  to 
have  not  only  a  general  education  but  also  a  more  thorough 
knowledge  of  history,  mathematics,  and  philosophy.  These  re- 
quirements did  not  go  into  effect,  however,  until  1813,  as  it  was 
considered  that  it  would  take  the  universities  three  years  to 
prepare  the  candidate  for  meeting  these  requirements. 

It  was  not  until  the  ordinance  of  1831  that  a  very  definite 
statement  of  the  requirements  was  made.  These  regulations 
gave  the  candidate  the  privilege  of  taking  one  of  four  different 
examinations,  according  to  the  kind  of  position  he  desired. 

1 i )  The  "  pro  f  acultate  docendi  "  examination,  which  was  the 
most  definitely  outlined  examination  and  was  considered  the  most 
important. 

(2)  The  "  pro  loco  "  examination,  which  was  not  found  prac- 
tical and  was  not  considered  legal  after  1866. 

(3)  The  "pro  ascensione  "  examination,  which  also  was  not 
open  to  candidates  after  1866. 

(4)  The  "  colloquia  pro  rectoratu  "  examination,  which  even 
to-day  is  one  of  the  examinations  for  which  a  candidate  may 
prepare. 


165]     The  Present  Teaching  of  Mathematics  in  Germany      99 

In  order  that  a  candidate  might  take  the  "  pro  facultate 
docendi  "  examination,  it  was  necessary  that  he  have  knowledge 
of  all  subjects,  i.e.,  languages  (Greek,  German,  Latin,  French, 
and  Hebrew),  mathematics,  physics,  German  history,  geography, 
mythology,  literature  of  the  Greeks  and  Romans,  philosophy, 
pedagogy. 

Having  passed  an  unconditioned  "  facultas  docendi,"  the  can- 
didate had  the  privilege  of  teaching  in  all  classes  of  the  Gym- 
nasium, provided  he  also  showed  a  special  knowledge  of  at  least 
two  of  the  ancient  languages,  his  own  language,  mathematics, 
nature  study  (botany,  mineralogy,  chemistry,  and  zoology),  his- 
tory, and  geography.  Wishing  to  teach  mathematics  in  the  Gym- 
nasium, a  candidate,  in  order  to  meet  the  requirements  of  the 
"  facultas  docendi,"  must  have  a  thorough  knowledge  of  the 
following  subjects: 

1 i )  Elementary  geometry  and  common  arithmetic  for  instruct- 
ing in  the  lower  classes  (which  are  the  same  as  our  fourth,  fifth, 
and  sixth  grades). 

(2)  Geometry,  including  both  plane  and  solid,  plane  trigonom- 
etry, and  common  arithmetic  for  teaching  in  the  middle  classes 
(which  correspond  to  our  seventh  and  eighth  grades,  and  the 
first  year  in  the  high  school). 

(3)  Higher  geometry,    analysis   of   infinity,   applications   of 
mathematics  to  astronomy  and  physics  (physics  and  mathematics 
being  correlated)  for  teaching  in  the  higher  classes  (which  cor- 
respond to  the  last  three  years  of  our  high  school  course). 

For  the  first  time,  one  finds  the  study  of  the  science  of  edu- 
cation emphasized.  Psychology,  logic,  and  philosophy,  including 
its  history  and  changes  since  the  time  of  Kant,  formed  an  inti- 
mate part  of  the  candidates'  studies.  Until  the  edict  of  1831 
mathematics  and  nature  study  had  been  correlated,  but  it  was 
then  decided  to  separate  the  two  subjects,  as  it  was  thought 
impossible  to  find  teachers  who  were  thoroughly  efficient  in 
both  departments.  Physics  was  considered  more  as  an  experi- 
mental study  than  a  mathematical  study.  The  candidate  was  re- 
quired to  have  a  general  knowledge  of  the  entire  subject,  to 
know  its  phenomena  in  nature  and  its  laws,  to  be  familiar  with 
the  apparatus  for  the  study  of  physics,  and  to  be  able  to  perform 


ioo  Teachers  College  Record  [166 

the  ordinary  experiments  in  class.  It  was  in  this  edict  that 
drawing  was  first  emphasized. 

The  second  and  third  examinations,  i.e.,  "pro  loco"  and  "pro 
ascensione,"  were  of  minor  importance.  The  aim  of  the  "  pro 
loco  "  examination  was  to  show  the  ability  of  the  candidate  for 
certain  positions  in  special  schools.  The  "pro  ascensione"  exam- 
ination was  for  the  teacher  who  wished  to  teach  in  the  higher 
schools. 

The  fourth  examination,  i.e.,  "  colloquia  pro  rectoratu  "  was  to 
determine  whether  a  candidate  for  the  rectorship  in  the  higher 
schools  was  capable  of  filling  the  position.  In  this  the  candidate 
was  examined  partly  in  Latin  and  partly  in  German  upon 
pedagogical  and  didactic  subjects.  One  is  no  longer  in  doubt  as 
to  one  of  the  reasons  why  Germany  has  been  so  successful  in 
building  up  a  school  system  of  such  great  force,  when  he  learns 
of  the  many  requirements  of  the  rector  of  the  higher  schools 
even  as  early  as  1831. 

Each  candidate  who  took  one  of  the  regular  examinations  was 
required  to  have  six  months  of  observation  either  in  a  Gym- 
nasium or  Realschule.  The  Oberlehrers  (teachers  in  the  higher 
classes)  especially  were  very  much  dissatisfied  with  the  recent 
ordinance.  They  were  greatly  agitated  over  the  question  of 
being  allowed  to  enter  a  pedagogical  seminary,  such  a  school  as 
one  finds  in  Teachers  College,  Columbia  University,  and  have 
the  work  done  there  count  towards  a  degree. 

Regardless  of  the  fact  that  mathematics  was  gradually  develop- 
ing in  the  universities,  its  progress  in  the  Gymnasien  was  much 
slower.  It  is  a  lamentable  fact  that  Gauss  of  the  University  of 
Gottingen,  at  the  time  of  the  passing  of  the  ordinance  of  1866, 
though  doing  so  much  to  promote  pure  mathematics  at  the  Uni- 
versity, took  no  pains  to  develop  in  his  students  a  knowledge  or 
interest  in  the  mathematics  of  the  secondary  schools.  His  atti- 
tude towards  the  advancement  of  mathematics  in  the  secondary 
schools  was  the  attitude  of  most  of  the  university  professors  of 
mathematics  at  that  time.  Indeed,  the  first  examiner  for  mathe- 
matics was  not  Gauss,  but  Thibaut,  the  professor  of  philosophy 
at  the  University  of  Gottingen.  The  advancement  of  the  schools 
was  entirely  in  the  hands  of  the  classics  instructors  who  tried  to 


167]     The  Present  Teaching  of  Mathematics  in  Germany     101 

retard  mathematics  as  much  as  possible.  It  was  almost  unheard 
of  at  that  time  for  a  mathematician  to  be  the  rector  of  a  higher 
school. 

The  ordinance  of  1866  was  one  for  which  the  teachers  had 
long  looked  and  hoped.  The  requirements  for  the  different 
classes  were  as  follows : 

I.  LOWEST   CLASSES:   plane   geometry,    stereometry,   common 
arithmetic,  algebra,  and  method  of  instruction  in  arithmetic. 

II.  MIDDLE  CLASSES:   plane  and   solid   geometry,    plane   and 
spherical  trigonometry,  algebra  up  to  equations  of  the  third  and 
fourth  degrees,  analytic  theory  of  straight  lines  and  planes  as 
applied  to  conies,  fundamental  operations  of  the  differential  and 
integral  calculus,  and  the  main  laws  of  statics. 

III.  HIGHER  CLASSES:    research  into  higher  geometry,  higher 
analysis,  and  analytic  mechanics.    This  work  was  to  be  the  basis 
for  independent  research.     The  candidate  in  mathematics  was 
required  to  be  examined  in  nature  study    (chemistry,  zoology, 
mineralogy,  and  botany).     The  candidate  in  nature  study  had 
to  meet  the  requirements  in  the  mathematics  of  the  middle  classes. 
The  main  point  emphasized  in  this  edict  was  that  the  candidate 
must  have  a  thorough  knowledge  of  the  technical  language  of 
the  specialized  subject. 

This  ordinance  of  1866  regulated  the  granting  of  three  grades 
of  certificates,  for  which  the  requirements  were  as  follows:  for 
the  first  grade  certificate,  mathematics  and  physics  for  the  highest 
classes,  and  nature  study  for  the  middle  classes ;  for  the  second 
grade  certificate,  a  capacity  for  mathematics,  physics,  and  one 
of  the  nature  studies  for  the  middle  classes ;  for  the  third  grade 
certificate  nothing  but  a  complete  knowledge  of  the  specialized 
subject.  The  teachers  were  very  much  opposed  to  these  certifi- 
cates. They  considered  it  detrimental  to  the  specialized  subject 
to  be  required  to  have  such  a  general  knowledge  in  order  to 
receive  a  first  grade  certificate. 

Another  ordinance  was  issued  in  1877,  but  the  mathematics 
requirements  remained  practically  the  same  with  the  exception 
that  a  candidate  for  the  higher  classes  was  required  to  know  more 
applied  mathematics.  The  fact  was  emphasized  that  the  candi- 
date should  have  a  clearer  insight  and  a  more  scientific  knowl- 


1O2  Teachers  College  Record  [168 

edge  than  was  necessary  for  the  teaching  of  the  subject.  It  was 
after  this  ordinance  was  issued  that  the  university  professors 
became  much  interested  in  the  development  of  mathematics  in 
the  higher  schools.  One  of  the  big  questions  that  was  being 
agitated  at  that  time,  as  it  is  to-day,  was  whether  the  stress 
should  be  laid  upon  the  method  of  instruction  or  upon  the 
scientific  knowledge. 

One  of  the  prominent  mathematicians  of  the  day  said  that  if 
the  student  were  allowed  to  devote  all  his  time  to  acquiring  a 
knowledge  of  his  science  instead  of  being  required  to  give  so 
much  of  it  to  methods  of  the  recitation,  he  would  have  a  firm 
foundation  upon  which  to  build,  and  very  soon  would  acquire 
the  method  by  means  of  experience. 

During  the  ten  years  following  the  regulations  of  1887,  Pro- 
fessor Klein  was  the  strongest  force  directing  the  tendencies 
of  mathematics  teaching  in  Prussia.  This  he  accomplished 
through  his  special  vacation  courses  for  teachers  which  were 
offered  in  the  universities.  Lectures  on  some  phase  of  mathe- 
matics teaching,  especially  on  applied  mathematics,  were  deliv- 
ered by  him  to  the  students  taking  these  courses,  and  to  the 
university  professors.  The  University  of  Gottingen  seems  to 
have  been  the  centre  of  his  influence,  establishing  courses  in 
science  and  mathematics,  and  encouraging  the  teaching  of 
applied  physics.  It  was  Professor  Klein's  enthusiasm  for 
reform  that  brought  him,  during  this  period,  to  the  United  States 
to  attend  a  congress  of  mathematicians  at  the  World's  Fair 
in  Chicago.  His  lectures  on  "  The  Present  State  of  Mathematics  " 
and  his  plea  for  an  international  union  of  mathematicians  con- 
tained the  ideas  which  he  brought  as  his  message  to  the  congress. 

By  1898  the  tendencies  in  mathematics  teaching  had  taken 
definite  form.  That  year  the  next  regulations  in  regard  to  teach- 
ers' examinations  were  passed  by  the  government  of  Prussia. 
The  most  important  of  the  new  requirements  were:  first,  that 
teachers  of  pure  mathematics  in  the  first  five  of  the  lower 
grades  should  have  a  thorough  knowledge  of  primary  mathe- 
matics, plane  geometry,  analytic  geometry,  integral  and  differ- 
ential calculus ;  second,  that  teachers  of  higher  mathematics  must 
be  masters  of  higher  geometry,  arithmetic  (theory  of  numbers), 


169]     The  Present  Teaching  of  Mathematics  in  Germany     103 

algebra,  higher  analysis  and  analytic  mechanics  in  order  to  be 
qualified  in  pure  mathematics;  third,  that  they  must  know 
applied  mathematics.  This  included  descriptive  geometry  as 
far  as  the  rules  of  central  projection,  mathematical  methods  for 
technical  mechanics  especially  graphic  statics,  and  higher  and 
lower  geodesy.  Another  regulation  was  that  mathematics  must 
always  be  connected  with  physics;  and  it  was  also  decided  that 
knowledge  of  astronomy  was  to  be  expected  of  a  candidate, 
though  no  formal  examination  should  be  required.  One  con- 
cession was  made;  namely,  that  the  candidate  found  deficient  in 
the  examination  could  be  permitted  to  teach  on  condition  that 
he  prove  himself  qualified  to  teach  one  of  the  higher  subjects 
or  two  of  the  lower  ones. 

These  regulations  of  1898  are  of  special  interest  because  they 
are  still  in  force,  no  radical  changes  having  been  made  by  the 
government.  However,  this  is  not  to  be  construed  as  a  sign  that 
such  a  standard  has  been  reached  that  no  further  effort  toward 
improvement  is  considered  necessary.  The  facts  show  that  the 
various  provinces  have  been  kept  busy  with  problems  demanding 
attention.  The  commissioners  who  make  up  the  examinations 
have  met  yearly  since  1900  to  discuss  changes  in  the  course  of 
study. 

In  1900  at  a  convention  of  university  professors  called  for  the 
purpose  of  discussing  some  measures  of  reform  for  the  higher 
schools  of  Berlin,  one  question  proposed  was,  What  can  be  done 
to  raise  the  standard  of  technical  and  applied  mathematics?  At 
the  same  conference  the  growing  interest  in  the  teaching  of  mathe- 
matics was  attested  by  the  presence  there  of  a  greater  number  of 
mathematics  teachers  than  had  been  present  at  any  previous  simi- 
lar gathering.  It  is  significant  too  that  the  subject  of  Professor 
Klein's  lecture  to  the  teachers  during  the  vacation  course  that 
year  was  technical  and  applied  mathematics.  Again,  in  1902, 
he  was  emphasizing  two  points  especially:  first,  that  candidates 
should  be  required  to  make  good  drawings  with  explanations 
of  solutions ;  second,  that  models  and  construction  work  were 
necessary  to  teach  technical  mathematics  properly.  Many  were 
interested  in  the  teaching  of  applied  mathematics.  There  was 
a  feeling  on  the  part  of  some  against  separating  it  from  the  pure 
mathematics  lest  the  candidate  shun  the  difficulties  of  the  pure 


IO4  Teachers  College  Record  [170 

mathematics  in  favor  of  the  applied  mathematics.  Others,  of 
whom  Studys  is  representative,  think  that  the  applied  mathe- 
matics is  of  value  only  when  connected  with  the  pure  mathe- 
matics. 

A  second  question  is  one  which  has  provoked  discussion  in 
our  country  as  well  as  in  Prussia:  What  shall  be  the  minimum 
of  general  culture  required  of  a  candidate  specializing  in  mathe- 
matics? Studys  insists  that  no  subject  except  mathematics  should 
have  any  claim.  He  dismisses  with  scorn  the  idea  of  rejecting 
a  candidate  who  has  passed  his  examinations  in  mathematics, 
merely  because  he  has  failed  in  religion ! 

How  have  the  new  tendencies  and  higher  standards  affected 
the  number  of  candidates  in  mathematics?  We  find  that  the 
number  has  varied  directly  with  the  difficulties  of  the  require- 
ments. In  1909  there  were  280  candidates,  to  25  in  1839.  Dur- 
ing the  years  from  1907  to  1909  the  increase  has  been  66^3  per 
cent.  Assuming  that  the  ratio  of  supply  and  demand  has  not 
varied  unreasonably,  these  statistics  show  a  marvelous  growth 
in  mathematics  teaching  in  Prussia. 

The  need  of  boards  of  examiners  arose  with  the  regulations 
of  1816  requiring  teachers  to  pass  certain  examinations.  There 
are  five  of  these  boards  in  all,  with  centres  of  organization  dis- 
tributed among  the  provinces.  Each  board  of  examiners  is 
subject  to  the  over-president  of  the  province  in  which  it  is 
located.  The  members  are  appointed  yearly  by  the  Minister  of 
Instruction  of  Prussia.  For  some  time  the  examiners  were 
without  exception  university  men,  but  gradually  public  school 
teachers  were  made  eligible.  However,  the  wisdom  of  this  step 
has  been  questioned  on  the  ground  that  the  public  school  men 
have  in  the  majority  of  cases  proved  incompetent.  It  is  claimed 
by  others  that  examinations  under  university  professors  have 
often  been  unfair,  that  the  public  school  men  are  better  fitted 
to  conduct  the  examinations  since  they  understand  better  the 
requirements  of  the  schools  and  are  less  bound  by  formulas. 
The  latter  opinion  has  evidently  prevailed.  In  some  of  the 
provinces  the  chairman  of  the  board  must  be  a  public  school 
teacher.  The  Minister  of  Instruction  has  spoken  in  favor  of 
retaining  them ;  and  the  educational  commission  of  1907  held 
that  examiners  should  all  be  public  school  men. 


171  ]     The  Present  Teaching  of  Mathematics  in  Germany     105 

The  message  for  us  in  this  report  is  written  large.  We  see 
that  time  and  energy,  and  above  all,  the  thought  of  some  of  the 
best  minds  of  Germany  have  entered  as  factors  in  determining 
these  standards  for  the  examinations  of  the  public  school  teach- 
ers. That  the  superior  excellence  of  their  schools  has  been  a 
result  of  all  this  care,  we  cannot  doubt.  If  we  are  to  keep  abreast 
of  the  present  movements  in  education  we  must  introduce  into 
our  schools  more  of  the  mathematics  demanded  by  the  practical 
problems  of  modern  life.  But  we  cannot  introduce  into  our 
schools  such  subjects  as  descriptive  geometry  and  differential 
calculus  until  we  have  teachers  for  them.  The  need  of  higher 
standards  for  our  schools  is  too  well  known  to  require  dis- 
cussion. The  question  with  us  is,  How  can  these  reforms  be 
accomplished?  We  cannot  look  to  a  centralized  government  to 
do  this,  nor  is  it  probable  that  this  method,  if  possible,  would 
be  for  the  best  interests  of  education  in  our  country.  The  start- 
ing point  for  this  reform  is  thought  to  be  in  our  larger  cities 
since  they  can  exert  such  a  wide  influence,  have  so  much  free- 
dom in  the  government  of  their  schools,  and  are  directly  re- 
sponsible for  the  welfare  of  so  large  a  part  of  our  population. 
A  few  of  them  have  already  started  the  ball  rolling.  New 
York,  for  instance,  now  requires  for  teachers  of  mathematics 
an  examination  in  trigonometry,  analytic  geometry,  and  the 
calculus,  beyond  the  subjects  included  in  the  course  of  study. 
With  a  view  to  encourage  this  spirit  of  progress,  and  to  arouse 
a  wholesome  rivalry  among  the  citifcs,  Commissioner  Claxton 
is  preparing  for  publication  statistics  from  reports  as  to  what 
each  of  the  cities  is  doing. 

But  whatever  machinery  may  be  put  in  motion  to  produce  these 
reforms  in  mathematics  teaching,  the  power  that  turns  the  wheels 
will  be  applied  by  institutions  typified  by  Teachers  College.  For 
from  these  institutions  should  go  out  teachers  who  are  informed 
not  only  as  to  how  mathematics  ought  to  be  taught  and  what 
mathematics  ought  to  be  taught,  but,  most  important  of  all,  who 
themselves  know  mathematics.  Therefore  we  consider  the  great- 
est good  which  it  is  within  the  power  of  these  institutions  to 
accomplish  is  by  wise  instruction  to  create  an  enthusiasm  for  a 
subject,  which  by  its  impetus  alone  will  carry  a  student  far 
beyond  the  limits  of  bare  "  requirements  "  for  teaching. 


CHAPTER  XV 
DESCRIPTIVE  GEOMETRY  IN  THE  REALSCHULEN1 

Louise  Eugenie  Harvey  and  Jessie  Mae  Reynolds 

The  material  for  this  report  was  collected  by  Dr.  Ziihlke  from 
three  sources, — the  official  regulations  on  the  subject  of  line- 
drawing  and  descriptive  geometry,  the  literature  in  the  depart- 
ment of  mathematics,  and  the  results  of  a  tour  of  investigation 
undertaken  during  the  previous  year.  This  tour  of  investigation 
carried  Dr.  Zuhlke  to  about  thirty  German  schools.  In  order  to 
compare  conditions  prevailing  among  them  with  those  which  had 
a  marked  influence  on  the  course  of  study  in  southern  Germany, 
he  also  visited  four  Austrian  Oberrealschulen. 

The  stress  laid  upon  the  subject  of  descriptive  geometry  in 
the  German  schools  and  the  claims  made  for  it  are  worthy  of 
consideration  by  us  who  have  hardly  ever  thought  of  the  subject 
as  one  of  general  importance.  The  report  discusses  the  place 
of  line-drawing  and  descriptive  geometry  in  the  mathematics 
curriculum  of  the  Realschulen,  giving  details  concerning  the 
number  of  hours  devoted  to  them  in  the  various  institutions  and 
sketching  some  of  the  plans  of  study.  Under  the  heading  of 
"  procedure  in  teaching,"  sundry  questions  of  method  are  treated. 
The  use  of  models  and  the  conditions  in  the  rooms  assigned  to 
drawing  are  briefly  discussed.  Finally  the  training  of  the 
teacher  is  considered. 

THE  PLACE  OF  LINE-DRAWING  IN  THE  CURRICULUM 
The  exact  designation  of  the  subject  does   not  seem  to  be 
quite  settled  in  Germany;  that  is,  there  is  some  question  as  to 
the   department   of   mathematics    to    which    it   belongs.     Line- 
drawing  and  descriptive  geometry  are  sometimes  called  concrete 

1  Der  Unterricht  im  Linearzeichnen  und  in  der  Darstellenden  Geometric 
an  den  deutschen  Realanstalten,  von  Dr.  Paul  Zuhlke,  Leipzig  und  Berlin, 
IQII. 

106  [172 


173]      The  Present  Teaching  of  Mathematics  in  Germany     107 

mathematics,  sometimes  practical  mathematics,  and  sometimes 
pure  mathematics.  Some  wish  to  make  descriptive  geometry 
part  of  the  course  in  line-drawing.  The  opinion  of  the  best 
educators  seems  to  be  that  skill  in  drawing  is  essential  in 
descriptive  geometry,  and  that  if  the  correct  result  is  to  be 
produced,  both  sides  of  the  subject,  the  theory  and  the  tech- 
nique, should  be  equally  considered.  Then  comes  the  question 
of  precedence:  Should  skill  in  drawing  be  required  first,  the 
theoretical  work  following,  or  should  theory  be  introduced  at 
the  beginning?  And  would  this  latter  method  spoil  much  of 
the  delight  in  the  beauty  of  the  work?  The  former  view,  that 
of  requiring  skill  in  drawing  first,  is  the  more  general  one. 

The  exact  relation  of  line-drawing  and  descriptive  geometry 
to  the  remaining  branches  of  study  has  been  determined  in  vari- 
ous parts  of  Germany.  In  the  Realschulen  of  Prussia,  line- 
drawing  was  a  separate  subject  from  1901  to  1909,  although 
it  was  cited  under  the  heading  of  "  drawing  "  in  the  official  pro- 
gram of  study.  In  1908  it  was  divided  into  two  sections,  mathe- 
matics and  drawing,  each  class  having  two  hours  of  instruction 
a  week  from  a  drawing  teacher  and  the  other  three  hours  from 
a  mathematics  teacher.  It  is  in  Wiirttemberg,  Saxony,  and  Baden 
that  line-drawing  and  descriptive  geometry  are  most  closely  con- 
nected with  mathematics,  and,  with  the  exception  of  five  hours' 
work  in  Wiirttemberg,  it  is  everywhere  obligatory.  In  the  Ober- 
realschulen  of  Hamburg  descriptive  geometry  is  elective  in  the 
following  sense:  four  hours  per  week  of  instruction  have  been 
prescribed,  two  of  which  are  designated  as  freehand  drawing  and 
two  as  descriptive  geometry.  Of  these  four  hours,  the  student 
must  elect  two,  but  it  is  left  to  him  whether  he  will  take  one 
from  each  group,  or  both  from  one.  Dr.  Ziihlke  comments  on 
this  arrangement  with  the  caustic  remark,  "  In  any  case  it 
can  be  imagined  that  a  graduate  of  a  Hamburg  Oberrealschule 
knows  nothing  of  descriptive  geometry." 

The  division  of  what  is  called  line-drawing,  and  especially 
its  distribution  among  different  teachers,  is  thought  unfortunate. 
The  administration  seems  to  sanction  the  arrangement,  however, 
judging  from  a  decree  in  which  it  is  stated  that  so  long  as  cer- 
tain difficulties  continue,  things  must  remain  as  they  are. 


io8 


Teachers  College  Record 


[174 


MATHEMATICS  CURRICULUM  IN  THE  REAL-SCHOOLS 
The  number  of  hours  devoted  to  line-drawing  and  descriptive 
geometry  will  perhaps  be  best  understood  from  the  following 
table  :x 


CLASS  IN  GKRMAJ 
CORRESPONDING 
TO  ABOUT: 

w  SCHOOL  

IV 

6th 
school 
year 

UIII 
7th 
school 
year 

O  III 

8th 
school 
year 

U  II 

9th 
school 
year 
1st 
H.  S. 

O  II 

10th 
school 
year 
2nd 
H.  S. 

UI 

llth 
school 
year 
3rd 
H.  S. 

0  I 

12th 
school 
year 
4th 
H.  S. 

IN     OUR    SCHOOLS 

Prussia 
Realschule             Line-drawing 
Oberrealschule    /  Line-drawing 
and    Realgymn.  {  Descriptive  geom. 

- 

2 

2 
2 

*toto 

2 

2 

* 

2 

* 

Bavaria 
Oberrealschule 
Realgymnasium 

Line-drawing 
Technical  drawing 
Descriptive  geom. 
Line-drawing 
Descriptive  sreom. 

1 

2 

* 
2 

* 
2 

* 

* 

1 

_ 

* 

• 

* 
2 

2 

Wurtltmberg 
Oberrealschule    (  Geometric  drawing 
and  Realgymn.  •)  Descriptive  geom. 
since  1910            [  Projective  drawing 

- 

* 

* 

1 

* 

* 

* 
2 

* 
2 

Saxony 
Oberrealschule     f 
Realgymnasium  j  Line-drawing 
Realg.  —  Abt.  of]  (descriptive  geom.) 
Reformgymn.      [ 

- 

- 

- 

1 

2 
2 

2 
2 
3 

2 
2 
3 

Baden 
Oberrealschule 
(now  Reahzym. 
in  Mannheim) 
Realgymnasium 
(Reform  plan) 

Instruction 
in 
representation 

- 

- 

- 

2 

2 
2 

2 
2 

2 
2 

Hesse 
Oberrealschule    /  Geometric 
Realgymnasium  \  drawing 

- 

- 

- 

1 

1 

2 

2 

Hamburg 
Oberrealschule 

Descriptive  geom. 

- 

- 

- 

- 

2 

2 

2 

AUace' 
Oberrealschule  . 

- 

* 

* 

* 

2 

2 

2 

The  most  liberal  obligatory  courses  are  those  of  the  Ober- 
realschulen  of  Baden,  which  devote  to  the  subject  two  hours  per 
week  during  the  last  four  years.  The  direct  opposite  to  these 
are  the  Realgymnasien  of  Hesse  which  evidently  do  not  consider 
the  subject  at  all.  The  course  as  an  optional  is  most  extensively 
pursued  in  Prussia,  where  two  hours  per  week  during  the  last 

1  In  this  table  an  asterisk  means  that  the  official  program  calls  for 
obligatory  instruction  in  the  respective  classes,  but  that  no  fixed  hours 
have  been  designated. 


J75]     The  Present  Teaching  of  Mathematics  in  Germany     109 


five  years  are  given  to  it.  Up  to  1904,  descriptive  geometry  was 
obligatory  in  the  seventh,  eighth,  and  ninth  classes  of  the  Ober- 
realschulen  of  Wiirttemberg,  where  two  hours  per  week  in  the 
seventh  year,  four  hours  in  the  eighth,  and  four  hours  in  the 
ninth  year,  were  required.  In  January,  1904,  the  course  was 
extended  one  hour  in  the  highest  class.  Through  an  arrange- 
ment made  in  April,  1910,  a  part  of  this  "  reform  "  was  discon- 
tinued, in  that  it  was  decided  that  in  the  obligatory  course  of 
the  eighth  and  ninth  classes,  descriptive  geometry  should  be 
treated  in  close  connection  with  analytic  geometry,  to  which  two 
hours  per  week  instead  of  three  were  now  given. 

Dr.  Ziihlke  states  that  a  comparison  of  the  number  of  hours 
with  those  of  the  Austrian  Oberrealschulen  leaves  much  to  be 
desired,  for  in  Austria  one  finds  a  total  of  fifteen  hours  per 
week  of  obligatory  work  in  geometric  drawing  and  descriptive 
geometry.  Indeed  the  study  of  solids  in  space  is  given  much 
more  time  in  Austria  than  in  Prussia.  The  proportion  of  work 
done  in  both  states  may  be  seen  from  the  following  table:1 


AUSTRIAN 

OBERREAI>SCHULEN 

I 

II 

III 

IV 

V 

VI 

VII 

Total 

Calculation  and  Math- 

matics     

3 

3 

3 

4 

4 

ISem.  4 

5 

26 

Geometric  drawing  and 
descriptive  geometry 

2 

2 

3 

3 

II  Sem.  3 
3 

2 

(25) 
15 

Freehand  drawing.  .  .  . 

4 

4 

4 

3 

3 

2 

3 

23 

PRUSSIAN 
OBERREALSCHULEX 

VI 

V 

IV 

UIII 

OIII 

UII 

Oil 

UI 

01 

Total 

Calculations     and     Mathe- 
matics   

5 

5 
2 

6 
2 

6 
2 

5 
2 

5 
2 

5 
2 

5 

2 

5 
2 

47 
16 

Freehand  drawing  

1  In  Austria  the  Oberrealschulen  are  seven-class  schools.  The  students 
of  the  lowest  class  must  be  ten  years  old, — about  as  old  as  the  students 
of  the  fifth  class  in  the  regular  German  Realschulen.  The  students  of 
the  highest  (7th)  class  are,  as  a  rule,  older  than  those  of  the  ninth  class 
in  the  German  schools,  as  a  result  of  the  "  Classification  Examinations  " 
in  Austria. 


no  Teachers  College  Record  [176 

The  southern  states  stand  somewhere  between  Prussia  and 
Austria  with  respect  to  the  time  devoted  to  the  subject.  For 
example,  Bavaria,  in  the  nine  years,  devotes  to  freehand  draw- 
ing and  line-drawing  a  total  of  twenty-six  hours  of  obligatory 
work,  as  compared  with  a  total  of  twenty-three  in  seven  years 
in  Austria  for  line-drawing  alone. 

Besides  the  statement  of  the  average  time  devoted  to  line- 
drawing  and  descriptive  geometry,  some  of  the  official  programs 
of  study  are  given.  That  of  1910  for  the  Prussian  Oberreal- 
schulen  and  Gymnasien  requires  the  following  courses : 

UII :  Perspective  drawing  of  solid  figures. 

UI:  Elements  of  descriptive  geometry. 

On  the  other  hand  the  following  courses  in  line-drawing  are 
elective : 

OIII :  Practice  in  the  use  of  the  compasses,  ruler  and  drawing- 
pen,  through  the  drawing  of  surface  designs  and  other  geometric 
forms. 

UII :  Geometric  description  of  single  solids  in  different  views, 
with  sections  and  the  development  of  surfaces. 

Oil  and  I:  Further  study  in  descriptive  geometry;  study  of 
shading  and  perspective. 

WORK  IN  THE  HUMANISTIC  GYMNASIEN 

The  report  contains,  in  an  appendix,  a  brief  statement  of  the 
work  in  space-perception  done  in  the  humanistic  Gymnasien.  The 
introduction  of  the  teaching  of  projection  in  the  course  of  study 
is  a  result  of  the  reform  movement.  In  Prussia  the  study  plan 
of  1901  made  a  great  advance  in  this  department,  in  so  far  as 
for  the  first  time  instruction  in  the  perspective  drawing  of 
solids  was  admitted.  Many  educators  energetically  favor  having 
projective  geometry  in  the  course  of  study  in  the  Gymnasien. 

Although  a  regulated  treatment  of  stereometric  constructions 
can  not  always  be  accomplished  in  the  instruction  in  mathe- 
matics, still  many  a  good  opening  for  that  kind  of  work  remains. 
For  example,  in  the  official  program  of  study  for  the  UII  of  the 
Gymnasium,  under  the  work  in  natural  science,  is  the  topic: 
"  Discussion  of  some  of  the  most  important  minerals  " ;  and, 
under  geography,  the  topic,  "  Elementary  mathematical  geogra- 


177]     The  Present  Teaching  of  Mathematics  in  Germany     in 

phy."  Both  of  these  departments  are  in  the  hands  of  the  teacher 
of  mathematics  in  the  UII  of  the  Gymnasium.  So  here  is  an  op- 
portunity for  a  very  easy  introductory  treatment  of  parallel  pro- 
jection inserted  in  the  instruction  in  mineralogy  and  geography. 
This  work  is  an  actual  fact  in  the  Kaiser  Friedrich  Gymnasium 
at  Frankfurt.  The  aim  of  the  work,  they  say,  is  to  introduce 
all  students,  and  not  merely  a  few,  to  geometric  and  especially 
to  stereometric  drawing.  The  technical  course  in  drawing  in. 
V  and  VI  gives  practice  in  using  ruler  and  compasses. 

UIII:  Exercises  in  perspective. 

OIII:  Drawing  of  simple  crystals,  cubes,  octahedrons,  etc. 

UII:  Analysis  of  minerals,  of  square  and  hexagonal  systems 
of  crystallization.  Extension  of  exercises  in  the  drawing  of 
crystals. 

UI:  Scientific  introduction  of  projection  in  solid  geometry. 

OI:  Cross-section  of  solids. 

The  teacher  of  mathematics  should  be  given  latitude  in  doing 
the  correlation  in  this  work. 

The  study  in  projection  may  be  discontinued  at  any  time,  for 
even  a  little  of  it  gives  many  advantages,  among  them  neatness 
and  elegance  in  drawing,  and  training  of  the  eye  and  hand.  Dr. 
Ziihlke  says  that  the  groundwork  of  instruction  in  projection 
and  the  elements  of  stereometric  drawing  must  become  obligatory 
in  all  Gymnasien  if  they  are  not  to  be  blamed  for  turning  out 

unpractical  youths. 

f 

PROCEDURE  IN  TEACHING 

Dr.  Ziihlke  believes  that  instruction  in  line-drawing  and 
descriptive  geometry  should  proceed,  as  in  every  study,  from  the 
near  to  the  remote.  Appeal  should  therefore  be  made  first  to 
the  sense-perception.  There  are,  also,  certain  fundamentals 
which  need  no  demonstration ;  certain  intuitions  upon  which  the 
teacher  can  easily  build.  Thus  the  student  should  be  guided 
slowly  but  constantly  from  intuitive  knowledge  to  acquired 
knowledge,  and,  to  this  end,  "  not  to  draw  mechanically,  but 
to  draw  intelligently,  must  be  the  highest  rule." 

A  teacher  who  begins  with  too  abstract  inquiries  tires  the 
student ;  and  weariness  is,  according  to  Herbart,  the  heaviest 


H2  Teachers  College  Record  [178 

wrong  of  any  course,  "  for  a  man  does  not  live  on  what  he  eats, 
but  on  what  he  digests."  However,  in  the  upper  grades,  the 
desire  for  abstract  thought  develops,  and  here, knowledge  must 
be  systematized.  Yet  Dr.  Ziihlke  warns  the  teacher  against  an- 
ticipating in  any  way  the  academic  course  of  study,  for  nothing 
is  more  pernicious  for  a  young  man  who  has  taken  descriptive 
geometry  in  the  high  school  than  the  consciousness,  when  he 
reaches  college  work,  that  he  has  already  had  it  all. 

In  considering  further  the  correct  relation  between  abstract  and 
concrete  knowledge,  shadow-picturing  was  given  as  an  example 
of  a  much  more  interesting  and  instructive  lesson  than  any 
arrangement  of  abstract  figures.  A  sound  pedagogue,  it  is  said, 
moves  continuously  here  and  there  between  theory  and  practice, 
making  a  sane  use  of  both. 

Descriptive  geometry  should  not  be  an  isolated  subject,  for  as 
such  it  is  unfruitful,  but  should  be  the  connecting  link  between 
pure  and  applied  mathematics.  In  beginning  the  subject,  in- 
struction should  proceed  along  the  lines  of  the  question-develop- 
ing method.  The  author  warns  against  too  much  explanation 
and  talking  on  the  part  of  the  teacher.  Papers  are  put  into  the 
hands  of  the  students  on  which  neither  the  method  of  solution, 
nor  the  result  is  given,  but  merely  the  hypothesis ;  or  a  complete 
drawing,  from  which  he  is  to  recognize  the  conditions  of  the 
problem,  is  given  him  for  brief  inspection. 

The  following  is  a  reproduction  of  an  actual  recitation  in  the 
UII  of  an  Oberrealschule  (which  corresponds  in  time  to  our 
first  year  in  high  school).  In  this  the  students  had  become 
accustomed  to  space  conceptions,  and  had  learned  to  see  the 
drawing  completed  before  it  was  actually  finished. 

TEACHER:  What  proposition  did  we  consider  in  the  last 
recitation  ? 

STUDENT:  We  considered  the  intersection  of  a  straight  line 
and  a  plane. 

TEACHER:     By  what  means  were  the  two  determined? 

STUDENT:  The  straight  line  was  given  by  its  projections; 
the  plane  by  its  traces. 

TEACHER:     Explain  the  method  of  solution. 


179]     The  Present  Teaching  of  Mathematics  in  Germany     113 

(One  student  draws  at  the  board;  several,  one  after  another, 
explain  the  solution.) 

TEACHER:  We  will  take  up  the  same  proposition  to-day,  but 
considered  from  other  data,  since  in  practice  a  plane  is  very 
seldom  given  by  its  traces.  By  what  other  means  can  a  plane 
be  determined? 

STUDENT:    A  plane  can  be  determined  by  three  points  not 


lying  in  a  straight  line,  or  by  a  straight  line  and  a  point  without, 
or  by  two  intersecting  straight  lines. 

TEACHER:  We  wish  three  points  for  our  plane.  Take,  on 
the  board,  A  ==  (2,  i,  3),  B  =  (4,  5,  6),  C  =  (6,  2,  4). 

(A  student  does  this.  The  board  is  covered  with  fine  gray 
lines,  which  cannot  be  distinguished  at  more  than  two  or  three 
meters  distance.  The  student  letters,  without  special  request, 
the  projections  of  the  separate  points  A',  A",  B',  B",  C,  C".) 

TEACHER:  Draw  also  the  projections  of  the  triangle  deter- 
mined by  the  points  A,  B,  C.  (This  is  done.)  Let  us  consider 


U4  Teachers  College  Record  [180 

for  a  moment  the  triangle  ABC  cut  out  of  this  copy-book 
cover,  which  is  blue  on  the  outside  and  white  inside.  Then 
would  like  or  different  colors  be  apparent  in  the  horizontal  and 
vertical  projections? 

STUDENT  :     Different  colors  would  be  seen. 

TEACHER:     Why  do  you  conclude  that? 

STUDENT:  The  triangle  A'  B'  C'  has  an  opposite  sense  of 
revolution  to  the  triangle  A"  B"  C". 

TEACHER  :  Hold  this  paper  triangle  about  as  the  figure  would 
indicate.  (A  student  does  this.) 

TEACHER:  Now  we  still  need  a  straight  line  g  which  we  will 
determine  by  two  of  its  points.  Take  PEEEF  (2,  5,  2),  Q  =E  (7, 
i,  7),  and  the  projection  of  the  straight  line  determined  by  them. 
(This  is  done;  the  points  P',  P" ,  Q',  Q"  are  marked  but  lightly, 
the  addition  of  the  letters  not  being  made  because  of  the  result- 
ing indistinctness  of  the  figure.  Only  g',  g"  are  drawn.) 

TEACHER  :  Now  hold  this  pointer  in  the  position  the  straight 
line  would  take  according  to  our  sketch.  (This  is  done.)  Now 
draw  in  all  the  data  in  your  note-books.  (The  students  do  this.) 

TEACHER:     How  will  we  now  proceed? 

STUDENT:  We  can,  as  in  the  last  recitation,  pass  through  the 
straight  line  g  an  auxiliary  plane  at  right  angles  to  one  of  the 
planes  of  projection  and  determine  the  intersection  of  this  auxil- 
iary plane  with  the  plane  of  the  triangle  ABC. 

TEACHER:  What  do  we  call  such  a  plane  which  is  at  right 
angles  to  one  of  the  planes  of  projection? 

STUDENT:     Such  a  plane  is  called  the  projecting  plane. 

TEACHER:  Should  we  choose,  in  this  case,  a  vertical  project- 
ing-plane,  or  a  horizontal  projecting-plane? 

STUDENT:  That  will  depend  upon  which  relation  gives  the 
more  favorable  intersection. 

TEACHER:  Now  determine  which  will  be  the  best  plan  here. 
(A  student  steps  forward,  holds  with  one  hand  the  paper  triangle 
in  the  correct  position,  designates  the  straight  line  g  by  a  knitting- 
needle  taken  from  the  teacher's  desk,  and  shows  the  positions  of 
the  two  projecting  planes  with  the  palm  of  his  hand.  Another 
student  designates  with  a  finger  the  direction  of  the  projecting 
plane  with  the  plane  of  the  triangle.  A  third  student  criticises 


i8ij     The  Present  Teaching  of  Mathematics  in  Germany     115 

the  result :  "  It  is  rather  immaterial  here  whether  the  auxiliary 
plane  is  chosen  at  right  angles  to  the  vertical  plane  or  to  the 
horizontal  plane.") 

TEACHER:  If  it  makes  scarcely  any  difference,  we  will  take 
it  otherwise  than  in  the  previous  recitation.  How  did  the  assist- 
ing plane  lie  there? 

STUDENT  :     At  right  angles  to  the  horizontal  plane. 

TEACHER:  Then  we  will  choose  this  time  a  vertical  project- 
ing plane.  But  first  hold  the  paper  triangle  and  the  straight  line 
in  the  correct  position.  (The  student  does  this.)  Indicate  now 
the  position  of  the  auxiliary  plane.  (This  is  done.)  Show  the 
intersection  of  the  auxiliary  plane  and  the  triangular  plane.  (This 
is  also  done.)  We  wish  to  determine  the  points  which  the  cut- 
ting-line has  in  common  with  the  perimeter  of  the  triangle. 
Proceed. 

STUDENT:     Its  vertical  projection  U"  V"  will  fall  on  g". 

TEACHER:     Why? 

STUDENT:  Everything  which  lies  in  the  auxiliary  plane  ap- 
pears in  the  vertical  projection  as  g".  The  intersecting  line  UV, 
lying  in  this  assisting  plane,  is  also  projected  vertically  in  g". 

TEACHER  :     Hence  what  points  are  now  known  to  us  ? 

STUDENT:     The  points  U"  V" . 

TEACHER:     How  are  U"  and  V"  found? 

STUDENT  :  With  the  help  of  the  proposition.  The  lower  and 
upper  projections  of  a  point  lie  on  a  common  perpendicular  to  the 
base-line.  (£/'  lies  on  a  |  to  base-line  passing  through  U") 
V  is  found  in  the  same  way  as  the  intersections  of  B'  C'  with 
the  perpendicular  to  the  base-line  passing  through  V '. 

TEACHER:     Therefore  what  have  we  established? 

STUDENT:     We  know  the  two  projections  of  the  cutting  line 
UV. 
'   TEACHER:     But  what  is  our  aim? 

STUDENT:  We  wish  to  ascertain  the  intersection  of  g  with 
the  plane  of  the  triangle. 

TEACHER  :     Then  what  remains  to  be  done  ? 

STUDENT  :     We  must  make  U  V  intersect  with  g. 

TEACHER  :  How  do  we  represent  the  intersection  S  of  the  two 
straight  lines? 


Ii6  Teachers  College  Record  [182 

STUDENT:  We  know,  first  of  all,  its  horizontal  projection  S', 
as  the  intersection  of  U'  V  and  g.  (Another  student  proceeds 
at  a  sign  from  the  teacher.)  The  upper  elevation  S"  lies,  in  the 
first  place,  on  the  perpendicular  to  the  base-line  through  S', 
and  secondly  on  g". 

TEACHER:  How  can  we  prove  that  we  have  constructed  the 
projection  of  5  accurately? 

STUDENT:  We  can  also  work  out  the  construction  with  the 
horizontal  projecting-plane  of  g. 

TEACHER:  That  would  be  too  detailed  for  us.  Who  can  sug- 
gest something  else?  (Pause.)  If  S  is  really  a  point  of  the 
triangular  plane,  so  must,  for  example,  B  S  lie  wholly  in  the 
plane  of  A  B  C.  How  can  we  prove  that? 

STUDENT:  We  must  find  out  whether  or  not  B  S  cuts  the 
side  A  C. 

TEACHER:     And  how  prove  that? 

STUDENT:  B'  S'  determines  on  A'  C'  a  point  and  B"  S"  on 
A"  C"  another  point.  We  must  prove  that  these  two  points 
lie  on  the  same  perpendicular  to  the  base-line. 

TEACHER:  Prove  it  from  our  drawing.  (A  student  does  this 
by  making  use  of  fine  lines.)  Reverse  the  blackboard.  Now  all 
draw  the  construction  just  considered  in  your  note-books.  (While 
this  takes  place,  the  teacher  passes  about  the  class-room  from 
one  student  to  another.  He  finds  a  student  who  has  made  the 
perpendicular  to  the  base-line  through  U'  cut  A'  B'  instead  of 
A'  C'.  The  teacher  brings  the  mistake  to  the  student's  notice, 
and  in  order  to  convince  him,  lets  him  turn  back  the  blackboard 
and  run  over  the  points  in  the  construction,  using  again  the 
paper  triangle  as  a  model.) 

TEACHER:  Have  you  all  completed  the  drawing?  We  must 
also  determine  what  parts  of  g  appear  as  visible  lines  in  the 
vertical  and  horizontal  projections  and  what  parts  do  not  so 
appear.  How  can  we  determine  this? 

STUDENT  :    With  the  help  of  our  paper  triangle. 

TEACHER:  But  we  do  not  wish  that.  We  will  use  the  draw- 
ing only. 

STUDENT  :  Then  we  could  perhaps  proceed  as  we  did  recently 
when  we  settled  the  proportion  of  visibility  in  the  case  of 
warped  lines. 


183]     The  Present  Teaching  of  Mathematics  in  Germany     117 

TEACHER:  Who  remembers  that?  Tell  us  how  you  would 
here  use  the  method  mentioned. 

STUDENT  :  I  first  seek  the  visibility  in  the  vertical  projection. 
There  is  one  and  only  one  vertically  projected  straight  line  which 
cuts  g  as  well  as  A  C  in  space.  It  is  the  straight  line  whose 
vertical  projection  is  the  point  U".  I  measure  down  this  line, 
and  perpendicularly  to  the  horizontal  projection.  The  horizontal 
projection  shews  me  that  I  come  first  upon  a  point  of  g,  then 
upon  a  point  of  A  C  (namely  C7).  Therefore  the  vertically  pro- 
jected part  of  g  that  we  are  concerned  with  lies  in  front  of  the 
triangular  plane.  I  draw  the  projection  of  the  part  U"  S"  as  a 
visible  line. 

(Another  student  does  the  same  thing  with  respect  to  the  hori- 
zontal projection.) 

TEACHER:     Draw  that  in  your  note-books. 

(A  signal  bell  announces  the  close  of  the  hour.)  You  may 
take  that  home  with  you.  As  your  lesson  for  the  next  hour 
solve  the  following  exercise:  A  triangle  is  determined  through 
the  co-ordinates  of  its  vertices  D  (2,  3,  4),  E  (5,  5,  2),  F  (7, 
2,  7).  There  is  a  straight  line  g  determined  by  the  two  points 
K  (2,  6,  6)  and  L  (7,  i,  i).  Find  the  intersection  of  the  straight 
line  g  with  the  plane  of  the  triangle  D  E  F. 

For  the  sake  of  clearness  it  is  well  that  work  with  ruler  and 
compasses  precede  freehand  drawing.  The  student  should  reach 
the  conclusion  that  the  figures  do  not  merely  represent  general 
objects  in  space,  as  if  they  were  drawn  for  the  proof  of  geomet- 
ric propositions,  but  that  their  results  should  conform  to  measure 
as  well  as  to  shape.  In  the  case  of  an  especially  gifted  pupil 
greater  freedom  may  be  allowed,  even  in  the  beginning. 

As  he  advances,  the  student  is  encouraged  to  work  out  inde- 
pendently more  complex  drawings.  Many  of  these  individual 
productions  show  such  ability  to  visualize  objects  in  space  that 
one  is  quite  justified  in  allowing  the  student  to  spend  his  time 
on  abstract,  unpractical  propositions.  Such  work  often  repays 
the  student  a  thousandfold.  The  author  apparently  enters  a  plea 
for  not  attempting  to  eliminate  all  abstract  knowledge  in  favor 
of  the  concrete. 

Opinions  differ  widely  concerning  the  technical  execution  of 


Ii8  Teachers  College  Record  [184 

drawing.  To  quote  from  two  writers :  "  The  pupils  make  a 
sketch  in  their  copybooks,  and  as  soon  as  they  have  completed 
the  construction  it  is  redrawn  on  a  clean  sheet  and  filled  in  with 
color.  The  pages  must  be  handed  in  clean,  and  the  drawings 
must  be  exact,  no  bungling  being  allowed.  If  necessary,  two  or 
even  three  copies  are  made."  Again,  "  The  drawing  must  be 
correct.  If  this  can  be  obtained  with  a  lead  pencil  so  much  the 
better.  A  laborious  coloring  of  the  work  is  a  useless  waste  of 
time,  an  amateurish  nothingness."  In  many  cases,  indeed,  the 
students  are  allowed  to  produce  technical,  freehand  drawings  on 
wrapping  paper  with  lead  and  colored  pencils.  But,  as  Groth- 
mann  says,  they  do  not  train  office  draughtsmen. 

Dr.  Zuhlke  considers  as  the  best  the  happy  medium.  All 
pupils  should  be  allowed  to  work  at  the  drawing-board,  the 
weakest  and  slowest  only  with  pencil,  the  better  and  faster  with 
drawing  pen,  the  most  advanced  being  permitted  the  use  of  water 
colors. 

By  some  it  has  been  suggested  that  colors  be  used  to  differen- 
tiate that  which  is  given  in  the  problem  from  that  which  is 
sought;  also  that  the  lines  which  are  to  assist  in  the  proof 
be  made  fainter  than  the  head-lines.  The  author  goes  on  to 
state  that  the  cherished  plan  of  drawing  the  forward  lines  heavier 
than  the  back,  is,  broadly  speaking,  a  condemnation  of  the  ex- 
istence of  parallel  perspective.  Moreover,  it  is  not  necessary 
to  make  clearly-drawn  figures  indistinct  by  going  over  them 
in  colors.  Ellipses  do  not  need  to  resemble  eggs.  Neither  does 
the  author  advocate  the  use  of  the  compasses  in  erecting  a  per- 
pendicular to  a  line ;  nor  the  laying  of  the  ruler  along  the  line 
and  the  pushing  of  the  triangle  along  it  until  the  vertex  falls  on 
the  given  point.  The  vertex  of  the  triangle  soon  becomes  worn 
and  the  drawing  consequently  becomes  inaccurate.  In  regard  to 
requiring  the  students  to  redraw  twice  or  even  three  times,  Dr. 
Zuhlke  advises  the  use  of  common  sense  and  a  little  discretion. 

THE  USE  OF  MODELS 

The  use  of  models  in  descriptive  geometry  is  discussed  at  some 
length.  There  are  many  steps,  it  is  said,  between  those  who 
expect  the  salvation  of  everything  from  the  use  of  models,  sep- 


185  J      The  Present  Teaching  of  Mathematics  in  Germany     119 

arating  themselves  from  these  aids  at  no  step  in  the  work,  and 
those  who  never  use  a  model  at  all.  By  the  use  of  such  helps 
the  student  is  liable  to  lose  his  proper  faculty  of  inner  sight,  the 
building  up  of  which  is  the  aim  of  the  course.  Models  are  espe- 
cially helpful  in  the  beginning,  while  the  student's  powers  of 
space-conception  are  still  weak,  but  it  is  recommended  that  they 
be  used  with  constant  discretion,  and  that  the  student  be  not 
permitted  to  have  the  model  in  hand  for  the  entire  work,  but 
only  until  the  difficulties  of  presentation  have  been  surmounted. 
Miiller  and  Presler  advocate  a  frugal  use  of  models,  saying  that 
the  value  of  instruction  in  projection  lies  in  the  fact  that  one 
learns  to  do  without  them.  Professor  Treutlein,  too,  is  quoted 
as  saying :  "  The  models  are  perhaps  there  to  make  themselves 
superfluous." 

In  regard  to  the  room  assigned  to  drawing,  conditions  should 
first  conform  to  the  requirements  of  school  hygiene.  For  line- 
drawing  and  descriptive  geometry  it  is  desirable  to  have  a  special 
room.  The  room  should  be  large  enough  to  give  to  each  student 
four  square  meters  of  floor  space.  It  should  be  placed  in  the 
top  story,  and  should  have  skylights  facing  the  north  as  usual. 
Since  this  kind  of  exact  drawing  strains  the  eyes  more  than 
freehand  drawing  it  should  be  given  in  the  morning.  This  has 
been  accomplished  in  most  schools. 

THE  TRAINING  OF  THE  TEACHER 

The  method  of  procedure  in  the  class  room  depends  in  an 
important  measure  upon  the  more  or  less  practical  training  of 
the  teacher.  A  drawing  teacher  not  trained  in  mathematics  will 
give  the  exercise  in  line-drawing  and  descriptive  geometry  quite 
differently  from  a  teacher  of  mathematics  unskilled  in  the  rules 
of  drawing:  whereas  one  will  scarcely  see  a  great  difference 
between  a  ready  drawer  with  the  capacity  for  mathematical 
judgment,  and  a  mathematician  perfected  in  the  rules  of  drawing. 
In  southern  Germany  descriptive  geometry  is  universally 
esteemed,  and  so  it  is  natural  that  special  work  in  the  subject 
is  given  in  the  training  of  the  future  teacher  of  mathematics. 

In  Bavaria,  according  to  the  regulations  of  May  26,  1873,  for 
examinations,  every  prospective  teacher  of  mathematics  was 
examined  in  descriptive  geometry,  with  the  exception  of  its 


I2O  Teachers  College  Record  [186 

application  to  perspective  and  shadow-construction.  The  present 
order  of  examinations,  regulated  by  the  decree  of  January  21, 
1895,  requires  a  four-hour  examination  paper  in  descriptive 
geometry.  The  student  can  acquire  the  requisite  knowledge  in 
the  technical  colleges,  which,  since  their  foundation  in  the  year 
1868,  are  equally  entitled  with  the  university,  to  train  the  candi- 
date for  teaching  mathematics  and  physics.  In  the  university  a 
suitable  course  in  descriptive  geometry  is  amply  provided  for. 

The  situation  in  Wiirttemberg  is  similar,  and  in  Baden  it  has 
been  looked  upon  for  a  long  time  as  the  duty  of  the  university 
to  train  the  future  teacher  of  mathematics  in  descriptive  geom- 
etry. In  the  University  of  Heidelberg  lectures  were  given  as 
early  as  1865  in  descriptive  geometry  with  respect  to  shadow- 
construction  and  perspective. 

Such  also  is  the  condition  in  central  Germany.  In  Saxony 
the  subject  was  given  in  the  Technical  College  of  Dresden  in 
1870,  and  in  the  university  in  1881,  when  Professor  Klein  was 
called  from  the  Technical  College  of  Munich  to  Leipzig  for  the 
newly  created  professorship  in  geometry.  Since  that  time,  Leip- 
zig, as  well  as  the  southern  universities,  has  given  much  atten- 
tion to  applied  mathematics. 

But  in  northern  Germany  the  situation  is  quite  different.  In 
the  year  1880  W.  Krumme  impressively  urged  that  careful  in- 
struction in  descriptive  geometry,  as  an  integral  part  of  the 
mathematical  training  of  the  pupils  of  colleges,  be  considered, 
and  added  the  bitter  complaint  that  the  universities  showed  only 
a  slight  comprehension  of  it. 

There  were  in  1908,  as  assistants  in  descriptive  geometry  in 
the  Technical  College  at  Berlin,  six  active  teachers  of  the  first 
class  of  whom  only  one  was  an  expert  in  applied  mathematics 

In  the  curriculum  of  almost  every  Gymnasium  and  Realschule 
is  to  be  found  the  subject  of  descriptive  geometry,  whereas  in 
the  United  States  it  is  practically  unheard  of  as  a  study  in 
the  secondary  schools.  No  doubt  its  prominence  in  the  German 
schools  is  partly  due  to  the  great  manufacturing  interests  of 
that  country.  Germany  is  far-sighted  enough  to  recognize  that 
the  development  in  her  masses  of  such  powers  as  that  of  being 
able  to  visualize  an  object  from  its  working  drawing  will  have 
great  influence  upon  her  industries. 


CHAPTER  XVI 


Eleanora    T.    Miller 

It  is  well  that  Professor  Smith  at  the  close  of  his  article  has 
added  a  word  of  encouragement  to  the  American  teachers,  be- 
cause we  cannot  read  these  reports  without  feeling  a  trifle  de- 
pressed by  the  apparent  preponderance  of  evidence  in  favor  of 
German  ideals  and  results  as  compared  with  the  ideals  which 
prevail  and  the  results  which  are  achieved  throughout  the  United 
States  as  a  whole. 

American  ideals  of  education  are  necessarily  different  from 
those  of  Germany  owing  to  the  great  difference  in  social  condi- 
tions, and  it  is  our  business  to  face  our  problems  frankly  and 
courageously  and  to  give  all  credit  where  it  is  due.  There  is 
one  matter  in  regard  to  the  difference  in  the  type  of  students 
found  in  the  secondary  schools  of  the  two  countries  which  has 
not  been  mentioned.  In  her  secondary  schools  Germany  educates 
a  more  or  less  picked  class  of  boys  and  girls  while  we  are 
attempting  to  educate  the  masses,  to/  raise  the  great  rank  and 
file  of  our  young  people  to  higher  and  higher  levels  of  intelli- 
gence. It  is  perfectly  futile  to  try  to  transplant  in  their  entirety 
foreign  educational  ideas  and  methods  to  our  own  land.  We 
ma)'  get  suggestions  from  Germany  but  the  applications  must 
be  determined  by  our  own  national  needs  and  our  own  national 
character.  One  may  learn  much  from  studying  the  details  of 
the  work  of  other  countries  and  it  is  to  be  hoped  that  the  reader 
of  these  reports  may  find  some  helpful  suggestions  from  a  careful 
study  of  the  experiences  of  the  German  teachers  of  mathematics. 

We  are  told  that  one  of  the  advantages  which  the  German 
schools  have  over  ours  is  in  the  length  of  the  school  day,  and 
we  wonder  whether  this  disadvantage  might  be  overcome.  Mod- 
ern psychological  experiments  tend  to  prove  that  what  we  ordin- 

187]  121 


122  Teachers  College  Record  [188 

arily  think  of  as  mental  fatigue  is  little  more  than  emotional 
repugnance  toward  the  work  in  hand,1  and  that  if  physical 
conditions  can  be  made  ideal  and  the  work  can  be  made  inter- 
esting enough  to  appear  to  the  student  to  be  worth  while,  the 
length  of  the  school  day  can  be  increased  without  any  danger 
of  impairing  the  health  of  the  students.  But,  of  course,  physical 
conditions  are  not  ideal  under  our  existing  system,  so  there  is 
no  use  in  our  trying  to  do  that  which  will  continue  to  be  im- 
possible so  long  as  we  can  not  perfectly  control  our  environment. 

No  one  will  deny  that  the  training  of  the  Gentian  teachers  is 
vastly  superior  to  that  of  our  own;  but  we  are  gradually 
raising  the  standards  in  this  country.  It  is  to  be  hoped  that 
the  ideals  of  this  generation  will  be  realized  by  the  next ;  and 
not  the  least  important  of  these  ideals  is  that  we  have  a  thor- 
oughly trained  and  enthusiastic  corps  of  teachers  who  are  not 
only  willing  but  anxious  to  take  advantage  of  every  opportunity 
that  is  offered  to  train  to  the  highest  point  of  efficiency  the 
scholar,  the  artisan,  and,  indeed,  every  individual  that  helps  to 
make  up  our  complex  social  fabric. 

It  is  not  our  purpose  here  to  map  out  a  definite  course  in 
mathematics  for  secondary  schools,  but  some  suggestions  ought 
to  be  offered  along  the  line  of  a  course  which  will  meet  the 
needs  of  the  different  classes  of  students,  and  Germany's  experi- 
ence can  in  some  respects  guide  us. 

To  put  all  students  through  the  same  mill  is  neither  practicable 
nor  possible,  no  matter  how  much  we  may  wish  to  have  every 
student  know  just  as  much  mathematics  as  we  are  able  to  teach 
him  during  his  high-school  course.  Our  greatest  difficulty  is  to 
be  met  in  the  regular  high  school  in  the  average  city  where  we 
must  meet  the  needs  of  the  boy  or  girl  who  is  preparing  for 
college,  the  boy  or  girl  who  must  drop  out  at  the  end  of  a  year 
or  two,  and  the  one  who  is  undecided  as  to  whether  or  not  he 
will  continue  work  beyond  that  of  the  secondary  school. 

Supposing  our  present  four-year  course  to  continue,  there 
seems  to  be  no  reason  why  all  students  of  the  usual  type  of 
high  school  should  not  be  given  the  same  kind  of  work  for 
two  years,  if  the  course  is  flexible  enough  to  admit  of  a  maximum 


1  E.  L.  Thorndike,  Mental  Fatigue,  Psychological  Review,  Nov.  1900. 


189]     The  Present  Teaching  of  Mathematics  in  Germany     123 

requirement  for  the  capable  student  and  a  minimum  requirement 
for  the  slow  one.  The  student  can  be  made  to  feel  that  mathe- 
matics is  worth  while  if  the  subject  is  properly  presented  by  a 
thoroughly  competent  teacher,  and  this  is  all  that  is  necessary  to 
hold  his  interest  and  call  forth  his  best  effort.  This  two  years  of 
work  should  include  algebra  and  plane  geometry,  and  some  trig- 
onometry in  connection  with  the  subject  of  similarity  of  triangles. 
Applications  to  business,  home  economics,  mechanics,  and  the 
various  other  sciences  can  be  introduced  in  such  a  way  as  to 
enlist  the  interest  of  both  boys  and  girls  of  different  bents  of 
mind. 

The  following  two  years'  work  might  be  required  for  those 
who  expect  to  continue  the  study  of  mathematics  in  some  higher 
institution,  and  elective  for  those  who  do  not  intend  to  go  to 
college  but  who  have  discovered  their  interest  in  the  work  pre- 
viously done.  As  electives  might  be  suggested  a  course  in  trig- 
onometry ;  a  course  in  mechanical  drawing  combined  with  descrip- 
tive geometry  which  should  be  a  genuine  mathematics  course 
and  not  a  "  snap  "  course  in  drawing,  open  to  everybne ;  a  course 
in  solid  geometry ;  a  course  in  advanced  algebra ;  and  a  course 
in  analytics  and  the  rudiments  of  the  calculus,  provided  there 
were  teachers  trained  to  present  these  subjects  properly.  The 
objection  that  the  presentation  of  such  subjects  as  analytics  and 
the  calculus  in  the  high  school  takes  off  the  keen  edge  of  the 
student's  interest  in  them  when  he  goes  to  college  is  a  reflection 
upon  the  teacher's  method,  and  is  not  an  objection  which  need 
be  taken  seriously. 

Our  vocational  schools  have  problems  which  are  more  easily 
solved  because  their  needs  are  not  so  varied  and  their  aims  are 
a  little  more  definite. 

The  educational  renaissance  through  which  we  are  passing 
to-day  is  being  felt  in  the  teaching  of  mathematics,  and  periods 
of  reform  are  always  more  or  less  disturbing.  We  are  to-day 
striving  to  make  our  concrete  procedure  come  up  to  and  fit 
in  with  the  needs  which  we  have  felt  for  some  time,  and  to 
adjust  our  mathematics  to  the  new  type  of  student  that  now 
comes  to  the  high  school.  When  the  readjustments  are  made, 
however,  and  we  settle  down  on  the  next  rung  of  the  ladder  we 


124  Teachers  College  Record  [190 

shall  probably  find  that  the  only  very  radical  changes  which  have 
been  brought  about  are  those  affecting  the  teachers  and  their 
methods,  and  that  the  subject  matter  will  be  much  the  same. 
What  we  most  desire  is  that  our  teachers  of  secondary  mathe- 
matics shall  be  thoroughly  familiar  with  their  field  far  beyond 
the  demands  of  the  curriculum  and  that  they  be  masters  of 
it  on  its  historical,  its  practical,  and  its  theoretical  side. 


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